diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..c2a5a1c --- /dev/null +++ b/.gitattributes @@ -0,0 +1 @@ +resources/* linguist-vendored diff --git a/resources/MayConcise/ConciseRevised.pdf b/resources/MayConcise/ConciseRevised.pdf new file mode 100644 index 0000000..34946b5 Binary files /dev/null and b/resources/MayConcise/ConciseRevised.pdf differ diff --git a/resources/MayConcise/ConciseRevised.tex b/resources/MayConcise/ConciseRevised.tex new file mode 100644 index 0000000..254b8d1 --- /dev/null +++ b/resources/MayConcise/ConciseRevised.tex @@ -0,0 +1,12340 @@ +\documentclass{amsbook} +\usepackage{amssymb, amsfonts, lacromay} +\usepackage[v2]{xy} + + + +%\makeindex + +%theoremstyle{plain} --- default +\newtheorem*{thm}{Theorem} +%\renewcommand{\thethm}{} +\newtheorem*{cor}{Corollary} +%\renewcommand{\thecor}{} +\newtheorem*{prop}{Proposition} +%\renewcommand{\theprop}{} +\newtheorem*{lem}{Lemma} +%\renewcommand{\thelem}{} + +\theoremstyle{definition} +\newtheorem*{defn}{Definition} +%\renewcommand{\thedefn}{} +\newtheorem*{defns}{Definitions} +%\renewcommand{\thedefns}{} +\newtheorem*{exmp}{Example} +%\renewcommand{\theexmp}{} +\newtheorem*{exmps}{Examples} +%\renewcommand{\theexmps}{} +\newtheorem*{con}{Construction} +%\renewcommand{\thecon}{} +\newtheorem*{notn}{Notation} +%\renewcommand{\thenotn}{} +\newtheorem*{notns}{Notations} +%\renewcommand{\thenotns}{} + +\theoremstyle{remark} +\newtheorem*{rem}{Remark} +%\renewcommand{\therem}{} +\newtheorem*{rems}{Remarks} +%\renewcommand{\therems}{} +\newtheorem*{warn}{Warning} +%\renewcommand{\thewarn}{} + +\def\cir{\mathaccent"7017} + +\newcommand{\0}{\phantom{1}} + +\makeatletter +\def\chapterrunhead#1#2#3{% + \def\@tempa{#3}% + \ifx\@empty\@tempa\else\uppercasenonmath\@tempa\@tempa\fi} +\makeatother + +%\renewcommand{\thechapter}{\arabic{chapter}} + +\title{A Concise Course in Algebraic Topology} + +\author{J. P. May} + +\begin{document} + +\renewcommand{\thepage}{\roman{page}} +\setcounter{page}{3} + +\maketitle + +\tableofcontents + +\clearpage + +\thispagestyle{empty} + +\chapter*{Introduction} + +\renewcommand{\thepage}{\arabic{page}} +\setcounter{page}{1} + +The first year graduate program in mathematics at the University of Chicago +consists of three three-quarter courses, in analysis, algebra, and topology. +The first two quarters of the topology sequence focus on manifold theory and +differential geometry, including differential forms and, usually, a glimpse +of de Rham cohomology. The third quarter focuses on algebraic topology. I have +been teaching the third quarter off and on since around 1970. Before that, the +topologists, including me, thought that it would be impossible to squeeze a +serious introduction to algebraic topology into a one quarter course, but we +were overruled by the analysts and algebraists, who felt that it was unacceptable +for graduate students to obtain their PhDs without having some contact with +algebraic topology. + +This raises a conundrum. A large number of students at +Chicago go into topology, algebraic and geometric. The introductory course +should lay the foundations for their later work, but it should also be +viable as an introduction to the subject suitable for those going into other +branches of mathematics. These notes reflect my efforts to organize the +foundations of algebraic topology in a way that caters to both pedagogical +goals. There are evident defects from both points of view. A treatment more +closely attuned to the needs of algebraic geometers and analysts would include +\v{C}ech cohomology on the one hand and de Rham cohomology and perhaps Morse +homology on the other. A treatment more closely attuned to the needs of +algebraic topologists would include spectral sequences and an array of +calculations with them. In the end, the overriding pedagogical goal has +been the introduction of basic ideas and methods of thought. + +Our understanding of the foundations of algebraic topology has undergone subtle but +serious changes since I began teaching this course. These changes reflect in part an +enormous internal development of algebraic topology over this period, one which +is largely unknown to most other mathematicians, even those working in such +closely related fields as geometric topology and algebraic geometry. Moreover, +this development is poorly reflected in the textbooks that have appeared over +this period. + +Let me give a small but technically important example. The study of +generalized homology and cohomology theories pervades modern algebraic topology. +These theories satisfy the excision axiom. One constructs most such theories homotopically, +by constructing representing objects called spectra, and one must then prove that +excision holds. There is a way to do this in general that is no more difficult +than the standard verification for singular homology and cohomology. I find this proof +far more conceptual and illuminating than the standard one even when specialized to +singular homology and cohomology. (It is based on the approximation of excisive triads +by weakly equivalent CW triads.) This should by now be a standard approach. However, +to the best of my knowledge, there exists no rigorous exposition of this approach in +the literature, at any level. + +More centrally, there now exist axiomatic treatments of large swaths of homotopy +theory based on Quillen's theory of closed model categories. While I do not think +that a first course should introduce such abstractions, I do think that the exposition +should give emphasis to those features that the axiomatic approach shows to be +fundamental. For example, this is one of the reasons, although by no means the only one, +that I have dealt with cofibrations, fibrations, and weak equivalences much more thoroughly +than is usual in an introductory course. + +Some parts of the theory are dealt with quite classically. The theory of fundamental +groups and covering spaces is one of the few parts of algebraic topology that has probably +reached definitive form, and it is well treated in many sources. Nevertheless, this material +is far too important to all branches of mathematics to be omitted from a first course. For +variety, I have made more use of the fundamental groupoid than in standard +treatments,\footnote{But see R. Brown's book cited in \S2 of the suggestions for further reading.} +and my use of it has some novel features. For conceptual interest, I have emphasized different +categorical ways of modeling the topological situation algebraically, and I have taken the +opportunity to introduce some ideas that are central to equivariant algebraic topology. + +Poincar\'e duality is also too fundamental to omit. There are more elegant ways to treat +this topic than the classical one given here, but I have preferred to give the theory in +a quick and standard fashion that reaches the desired conclusions in an economical way. +Thus here I have not presented the truly modern approach that applies to generalized homology +and cohomology theories.\footnote{That approach derives Poincar\'e duality as a consequence of +Spanier-Whitehead and Atiyah duality, via the Thom isomorphism for oriented vector bundles.} + +The reader is warned that this book is not designed as a textbook, although it could +be used as one in exceptionally strong graduate programs. Even then, it would be +impossible to cover all of the material in detail in a quarter, or even in a year. +There are sections that should be omitted on a first reading and others that are +intended to whet the student's appetite for further developments. In practice, when +teaching, my lectures are regularly interrupted by (purposeful) digressions, most often +directly prompted by the questions of students. These introduce more advanced topics that +are not part of the formal introductory course: cohomology operations, characteristic +classes, $K$-theory, cobordism, etc., are often first introduced earlier in the lectures +than a linear development of the subject would dictate. + +These digressions have been expanded and written up here as sketches +without complete proofs, in a logically coherent order, in the last four chapters. These are +topics that I feel must be introduced in some fashion in any serious graduate level introduction to +algebraic topology. A defect of nearly all existing texts is that they do not go far enough into +the subject to give a feel for really substantial applications: the reader sees +spheres and projective spaces, maybe lens spaces, and applications accessible with knowledge of +the homology and cohomology of such spaces. That is not enough to give a real feeling for the +subject. I am aware that this treatment suffers the same defect, at least before its sketchy +last chapters. + +Most chapters end with a set of problems. Most of these ask for computations and applications +based on the material in the text, some extend the theory and introduce further concepts, some +ask the reader to furnish or complete proofs omitted in the text, and some are essay questions +which implicitly ask the reader to seek answers in other sources. Problems marked $*$ +are more difficult or more peripheral to the main ideas. Most of these problems are +included in the weekly problem sets that are an integral part of the course at Chicago. In fact, +doing the problems is the heart of the course. (There are no exams and no grades; students are +strongly encouraged to work together, and more work is assigned than a student can reasonably be +expected to complete working alone.) {\em The reader is urged to try most of the problems: this is +the way to learn the material}. The lectures focus on the ideas; their assimilation requires +more calculational examples and applications than are included in the text. + +I have ended with a brief and idiosyncratic guide to the literature for the +reader interested in going further in algebraic topology. + +These notes have evolved over many years, and I claim no originality for most of the material. +In particular, many of the problems, especially in the more classical chapters, are the same as, +or are variants of, problems that appear in other texts. Perhaps this is unavoidable: interesting +problems that are doable at an early stage of the development are few and far between. I am +especially aware of my debts to earlier texts by Massey, Greenberg and Harper, Dold, and Gray. + +I am very grateful to John Greenlees for his careful reading and suggestions, especially +of the last three chapters. I am also grateful to Igor Kriz for his suggestions +and for trying out the book at the University of Michigan. By far my greatest debt, +a cumulative one, is to several generations of students, far too numerous to name. +They have caught countless infelicities and outright blunders, and they have contributed +quite a few of the details. You know who you are. Thank you. + +\clearpage + +\thispagestyle{empty} + +\chapter{The fundamental group and some of its applications} + +We introduce algebraic topology with a quick treatment of standard material about the +fundamental groups of spaces, embedded in a geodesic proof of the Brouwer fixed point +theorem and the fundamental theorem of algebra. + +\section{What is algebraic topology?} + +A topological space $X$ is a set in which there is a notion of nearness of +points. Precisely, there is given a collection of ``open'' subsets of $X$ +which is closed under finite intersections and arbitrary unions. It suffices +to think of metric spaces. In that case, the open sets are the arbitrary unions +of finite intersections of neighborhoods $U_{\epz}(x) = \{ y| d(x,y) < \epz \}$. + +A function $p: X\rtarr Y$ is continuous if it takes nearby points to +nearby points. Precisely, $p^{-1}(U)$ is open if $U$ is open. If $X$ and +$Y$ are metric spaces, this means that, for any $x\in X$ and $\epz>0$, there +exists $\de>0$ such that $p(U_{\de}(x))\subset U_{\epz}(p(x))$. + +Algebraic topology assigns discrete algebraic invariants to topological +spaces and continuous maps. More narrowly, one wants the algebra to be +invariant with respect to continuous deformations of the topology. +Typically, one associates a group $A(X)$ to a space $X$ and a homomorphism +$A(p): A(X)\rtarr A(Y)$ to a map $p: X\rtarr Y$; one usually writes $A(p) = p_*$. + +A ``homotopy'' $h:p\htp q$\index{homotopy} between maps $p,q: X\rtarr Y$ is a continuous +map $h: X\times I \rtarr Y$ such that $h(x,0)=p(x)$ and $h(x,1)=q(x)$, +where $I$ is the unit interval $[0,1]$. We usually want $p_*=q_*$ if +$p\htp q$, or some invariance property close to this. + +In oversimplified outline, the way homotopy theory works is roughly this. +\begin{enumerate} +\item One defines some algebraic construction $A$ and proves that it is +suitably homotopy invariant. +\item One computes $A$ on suitable spaces and maps. +\item One takes the problem to be solved and deforms it to the point that +step 2 can be used to solve it. +\end{enumerate} + +The further one goes in the subject, the more elaborate become the constructions +$A$ and the more horrendous become the relevant calculational techniques. This +chapter will give a totally self-contained paradigmatic illustration of the basic +philosophy. Our construction $A$ will be the ``fundamental group.'' We will calculate +$A$ on the circle $S^1$ and on some maps from $S^1$ to itself. We will then use the +computation to prove the ``Brouwer fixed point theorem'' and the ``fundamental +theorem of algebra.'' + +\section{The fundamental group} + +Let $X$ be a space. Two paths $f,g: I\rtarr X$ from $x$ to $y$ are +equivalent\index{equivalent!paths} +if they are homotopic through paths from $x$ to $y$. That is, there must exist +a homotopy $h:I\times I\rtarr X$ such that +$$h(s,0)=f(s), \ \ h(s,1)=g(s), \ \ h(0,t)=x, \ \tand \ h(1,t)=y$$ +for all $s,t\in I$. Write $[f]$ for the equivalence class of $f$. We say that +$f$ is a loop\index{loop} if $f(0)=f(1)$. Define $\pi_1(X,x)$\index{fundamental group|(} to +be the set of equivalence +classes of loops that start and end at $x$. + +For paths $f:x\to y$ and $g:y\to z$, define $g\cdot f$ to be the path +obtained by traversing first $f$ and then $g$, going twice as fast on each: +$$(g\cdot f)(s)= +\begin{cases} +f(2s) & \text{if $0\leq s\leq 1/2$} \\ +g(2s-1) & \text{if $1/2\leq s\leq 1$}. +\end{cases} +$$ +Define $f^{-1}$ to be $f$ traversed the other way around: $f^{-1}(s)=f(1-s)$. +Define $c_x$ to be the constant loop at $x$: $c_x(s)=x$. Composition of paths +passes to equivalence classes via $[g][f]=[g\cdot f]$. It is easy to check that +this is well defined. Moreover, after passage to equivalence classes, this +composition becomes associative and unital. It is easy enough to write down +explicit formulas for the relevant homotopies. It is more illuminating to draw a +picture of the domain squares and to indicate schematically how the +homotopies are to behave on it. In the following, we assume given paths +$$f:x\to y,\ \ \ g:y\to z,\ \tand \ \ h: z\to w.$$ + +$$h\cdot (g\cdot f) \htp (h\cdot g)\cdot f$$ +$$\diagram +\xline[0,4]^<(0.25)f ^<(0.65)g ^<(0.85)h \xline[4,0]_{c_x} + & & \xline[4,-1] & \xline[4,-1] & \xline[4,0]^{c_w} \\ + & & & & \\ + & & & & \\ + & & & & \\ +\xline[0,4]_<(0.15)f _<(0.35)g _<(0.75)h & & & & \\ +\enddiagram$$ + +$$f\cdot c_x \htp f \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c_y\cdot f\htp f$$ +$$\diagram +\rrline^f \ddline_{c_x} \ddrline & & \ddline^{c_y} & + & \rrline^f \ddline_{c_x} & & \ddline^{c_y} \ddlline\\ +& & & & & & \\ +\rrline_<(0.25){c_x} _<(0.75)f & & & & \rrline_<(0.25){f} _<(0.75){c_y} & & \\ +\enddiagram$$ +Moreover, $[f^{-1}\cdot f] = [c_x]$ and $[f\cdot f^{-1}]=[c_y]$. For the first, +we have the following schematic picture and corresponding formula. In the +schematic picture, +$$f_t=f|[0,t] \ \ \ \tand \ \ \ f^{-1}_t=f^{-1}|[1-t,1].$$ +$$\diagram +\xline[0,4]^<(0.25){f} ^<(0.75){f^{-1}} \xline[4,0]_{c_x} +& & \xline[4,-2] \xline[4,2] & & \xline[4,0]^{c_x} \\ +& & & & \\ +\xdotted[0,4]^<(0.15){f_t} ^<(0.5){c_{f(t)}} ^<(0.85){f^{-1}_t} & & & & \\ +& & & & \\ +\xline[0,4]_{c_x} & & & &\\ +\enddiagram$$ +$$h(s,t) = +\begin{cases} +f(2s) & \text{if $0\leq s\leq t/2$} \\ +f(t) & \text{if $t/2\leq s\leq 1-t/2$} \\ +f(2-2s) & \text{if $1-t/2\leq s\leq 1$.} +\end{cases} +$$ +We conclude that $\pi_1(X,x)$ is a group with identity element $e=[c_x]$ and +inverse elements $[f]^{-1}=[f^{-1}]$. It is called the fundamental +group\index{fundamental group|)} of +$X$, or the first homotopy group of $X$. There are higher homotopy groups $\pi_n(X,x)$ +defined in terms of maps $S^n\rtarr X$. We will get to them later. + +\section{Dependence on the basepoint} + +For a path $a: x\to y$, define $\ga [a]: \pi_1(X,x)\rtarr \pi_1(X,y)$ +by $\ga [a][f]=[a\cdot f\cdot a^{-1}]$. It is easy to check that $\ga [a]$ +depends only on the equivalence class of $a$ and is a homomorphism of groups. +For a path $b:y\to z$, we see that $\ga [b\cdot a]=\ga [b]\com \ga [a]$. It +follows that $\ga [a]$ is an isomorphism with inverse $\ga [a^{-1}]$. For +a path $b:y\to x$, we have $\ga[b\cdot a][f] = [b\cdot a][f][(b\cdot a)^{-1}]$. +If the group $\pi_1(X,x)$ happens to be Abelian, which may or may not be the +case, then this is just $[f]$. By taking $b=(a')^{-1}$ for another path +$a': x\to y$, we see that, when $\pi_1(X,x)$ is Abelian, $\ga [a]$ is +independent of the choice of the path class $[a]$. Thus, in this case, we have +a canonical way to identify $\pi_1(X,x)$ with $\pi_1(X,y)$. + +\section{Homotopy invariance} + +For a map $p: X\rtarr Y$, define $p_*: \pi_1(X,x)\rtarr \pi_1(Y,p(x))$ by +$p_*[f] = [p\com f]$, where $p\com f$ is the composite of $p$ with the loop +$f:I\rtarr X$. Clearly $p_*$ is a homomorphism. The identity map +$\id: X\rtarr X$ induces the identity homomorphism. For a map +$q: Y\rtarr Z$, $q_*\com p_* = (q\com p)_*$. + +Now suppose given two maps $p,q: X\rtarr Y$ and a homotopy $h: p\htp q$. +We would like to conclude that $p_*=q_*$, but this doesn't quite make sense +because homotopies needn't respect basepoints. However, the homotopy $h$ +determines the path $a: p(x)\to q(x)$ specified by $a(t)=h(x,t)$, and the +next best thing happens. + +\begin{prop} +The following diagram is commutative: +$$\diagram +& \pi_1(X,x) \dlto_{p_*} \drto^{q_*} & \\ +\pi_1(Y,p(x)) \rrto_{\ga [a]} & & \pi_1(Y,q(x)). \\ +\enddiagram$$ +\end{prop} +\begin{proof} +Let $f: I\rtarr X$ be a loop at $x$. We must show that $q\com f$ is +equivalent to $a\cdot (p\com f)\cdot a^{-1}$. It is easy to check that this is +equivalent to showing that $c_{p(x)}$ is equivalent to +$a^{-1}\cdot (q\com f)^{-1}\cdot a\cdot (p\com f)$. +Define $j: I\times I\rtarr Y$ by $j(s,t)=h(f(s),t)$. Then +$$j(s,0)=(p\com f)(s), \ \ \ j(s,1)=(q\com f)(s), \ \tand \ j(0,t)=a(t)=j(1,t).$$ +Note that $j(0,0)=p(x)$. Schematically, on the boundary of the square, $j$ is +$$\diagram +\rrline^{q\com f}|\tip & &\\ +& &\\ +\uuline^{a}|\tip \rrline_{p\com f}|\tip & & \uuline_{a}|\tip \\ +\enddiagram$$ +Thus, going counterclockwise around the boundary starting at $(0,0)$, we +traverse $a^{-1}\cdot (q\com f)^{-1}\cdot a\cdot (p\com f)$. The map $j$ induces +a homotopy through loops between this composite and $c_{p(x)}$. Explicitly, +a homotopy $k$ is given by $k(s,t)=j(r_t(s))$, where $r_t: I\rtarr I\times I$ +maps successive quarter intervals linearly onto the edges of the bottom left +subsquare of $I\times I$ with edges of length $t$, starting at $(0,0)$: +$$\diagram +\rrline \ddline|<(0.75)\tip & & \ddline \\ +& \lline|\tip & \\ +\rrline|<(0.25)\tip & \uline|\tip & \\ +\enddiagram$$ +\end{proof} + +\section{Calculations: $\pi_1(\bR )=0$ and $\pi_1(S^1)=\bZ$} + +Our first calculation is rather trivial. We take the origin $0$ as a convenient +basepoint for the real line $\bR$. + +\begin{lem} +$\pi_1(\bR,0)=0$. +\end{lem} +\begin{proof} +Define $k: \bR\times I\rtarr \bR$ by $k(s,t)=(1-t)s$. Then $k$ is a homotopy +from the identity to the constant map at $0$. For a loop $f: I\rtarr \bR$ at +$0$, define $h(s,t)=k(f(s),t)$. The homotopy $h$ shows that $f$ is equivalent +to $c_0$. +\end{proof} + +Consider the circle $S^1$ to be the set of complex numbers $x=y+iz$ of norm $1$, +$y^2+z^2=1$. Observe that $S^1$ is a group under multiplication of complex +numbers. It is a topological group: multiplication is a continuous function.\index{topological +group} +We take the identity element $1$ as a convenient basepoint for $S^1$. + +\begin{thm} +$\pi_1(S^1,1)\iso \bZ$. +\end{thm} +\begin{proof} +For each integer $n$, define a loop $f_n$ in $S^1$ by $f_n(s)=e^{2\pi ins}$. This is +the composite of the map $I\rtarr S^1$ that sends $s$ to $e^{2\pi is}$ and the $n$th +power map on $S^1$; if we identify the boundary points $0$ and $1$ of $I$, then the +first map induces the evident identification of $I/\pa I$ with $S^1$. It is easy +to check that $[f_m][f_n]=[f_{m+n}]$, and we define a homomorphism +$i:\bZ\rtarr \pi_1(S^1,1)$ by $i(n)=[f_n]$. We claim that $i$ is +an isomorphism. The idea of the proof is to use the +fact that, locally, $S^1$ looks just like $\bR$. + +Define $p: \bR\rtarr S^1$ by +$p(s)=e^{2\pi i s}$. Observe that $p$ wraps each interval $[n,n+1]$ around +the circle, starting at $1$ and going counterclockwise. Since the +exponential function converts addition to multiplication, we easily check that +$f_n=p\com \tilde{f}_n$, where $\tilde{f}_n$ is the path in $\bR$ defined by +$\tilde{f}_n(s)=sn$. + +This lifting of paths works generally. For any path +$f:I\rtarr S^1$ with $f(0)=1$, there is a unique path $\tilde{f}:I\rtarr\bR$ +such that $\tilde{f}(0)=0$ and $p\com \tilde{f}=f$. To see this, observe that +the inverse image in $\bR$ of any small connected neighborhood in $S^1$ is a +disjoint union of a copy of that neighborhood contained in each interval $(r+n,r+n+1)$ for +some $r\in [0,1)$. Using the fact that $I$ is compact, we see that we can +subdivide $I$ into finitely many closed subintervals such that $f$ carries each +subinterval into one of these small connected neighborhoods. Now, proceeding +subinterval by subinterval, we obtain the required unique lifting of $f$ by +observing that the lifting on each subinterval is uniquely determined by the +lifting of its initial point. + +Define a function $j:\pi_1(S^1,1)\rtarr \bZ$ by +$j[f]=\tilde{f}(1)$, the endpoint of the lifted path. This is an integer +since $p(\tilde{f}(1)) =1$. We must show that this integer is independent +of the choice of $f$ in its path class $[f]$. In fact, if we have a homotopy +$h: f\htp g$ through loops at $1$, then the homotopy lifts uniquely to a +homotopy $\tilde{h}: I\times I\rtarr \bR$ such that $\tilde{h}(0,0)=0$ and +$p\com \tilde{h}=h$. The argument is just the same as for $\tilde{f}$: we use +the fact that $I\times I$ is compact to subdivide it into finitely many subsquares +such that $h$ carries each into a small connected neighborhood in $S^1$. We then +construct the unique lift $\tilde{h}$ by proceeding subsquare by subsquare, starting +at the lower left, say, and proceeding upward one row of squares at a time. By +the uniqueness of lifts of paths, which works just as well for paths with any +starting point, $c(t)=\tilde{h}(0,t)$ and $d(t)=\tilde{h}(1,t)$ specify +constant paths since $h(0,t)=1$ and $h(1,t)=1$ for all $t$. Clearly $c$ is +constant at $0$, so, again by the uniqueness of lifts of paths, we must have +$$\tilde{f}(s)=\tilde{h}(s,0) \ \ \ \tand \ \ \ \tilde{g}(s)=\tilde{h}(s,1).$$ +But then our second constant path $d$ starts at $\tilde{f}(1)$ and ends at +$\tilde{g}(1)$. + +Since $j[f_n]=n$ by our explicit formula for $\tilde{f}_n$, +the composite $j\com i:\bZ\rtarr \bZ$ is the identity. It suffices to +check that the function $j$ is one-to-one, since then both $i$ and $j$ will be +one-to-one and onto. Thus suppose that $j[f]=j[g]$. This means that +$\tilde{f}(1)=\tilde{g}(1)$. Therefore $\tilde{g}^{-1}\cdot \tilde{f}$ is a +loop at $0$ in $\bR$. By the lemma, $[\tilde{g}^{-1}\cdot \tilde{f}]=[c_0]$. +It follows upon application of $p_*$ that +$$[g^{-1}][f]=[g^{-1}\cdot f]=[c_1].$$ +Therefore $[f]=[g]$ and the proof is complete. +\end{proof} + +\section{The Brouwer fixed point theorem} + +Let $D^2$ be the unit disk $\{ y+iz| y^2+z^2\leq 1\}$. Its boundary is $S^1$, +and we let $i: S^1\rtarr D^2$ be the inclusion. Exactly as for $\bR$, we see that +$\pi_1(D^2)=0$ for any choice of basepoint. + +\begin{prop} +There is no continuous map $r: D^2\rtarr S^1$ such that $r\com i=\id$. +\end{prop} +\begin{proof} +If there were such a map $r$, then the composite homomorphism +$$\diagram +\pi_1(S^1,1)\rto^<(0.2){i_*} & \pi_1(D^2,1) \rto^<(0.2){r_*} & \pi_1(S^1,1) \\ +\enddiagram$$ +would be the identity. Since the identity homomorphism of $\bZ$ does not factor +through the zero group, this is impossible. +\end{proof} + +\begin{thm}[Brouwer fixed point theorem]\index{Brouwer fixed point theorem} +Any continuous map +$$f: D^2\rtarr D^2$$ +has a fixed point. +\end{thm} +\begin{proof} +Suppose that $f(x)\neq x$ for all $x$. Define $r(x)\in S^1$ to be the +intersection with $S^1$ of the ray that starts at $f(x)$ and passes through $x$. +Certainly $r(x)=x$ if $x\in S^1$. By writing an equation for $r$ in terms of $f$, +we see that $r$ is continuous. This contradicts the proposition. +\end{proof} + +\section{The fundamental theorem of algebra} + +Let $\io\in\pi_1(S^1,1)$ be a generator. For a map $f: S^1\rtarr S^1$, define +an integer $\text{deg}(f)$\index{degree of a map} by letting the composite +$$\diagram +\pi_1(S^1,1)\rto^<(0.2){f_*} & \pi_1(S^1,f(1)) \rto^{\ga [a]} & \pi_1(S^1,1) \\ +\enddiagram$$ +send $\io$ to $\text{deg}(f)\io$. Here $a$ is any path $f(1)\to 1$; $\ga [a]$ is +independent of the choice of $[a]$ since $\pi_1(S^1,1)$ is Abelian. If $f\htp g$, +then $\text{deg}(f)=\text{deg}(g)$ by our homotopy invariance diagram and this independence of +the choice of path. Conversely, our calculation of $\pi_1(S^1,1)$ implies that if +$\text{deg}(f)=\text{deg}(g)$, then $f\htp g$, but we will not need that for the moment. It is +clear that $\text{deg}(f)=0$ if $f$ is the constant map at some point. It is also clear that if +$f_n(x)=x^n$, then $\text{deg}(f_n)=n$: we built that fact into our proof that $\pi_1(S^1,1)=\bZ$. + +\begin{thm}[Fundamental theorem of algebra]\index{fundamental theorem!of algebra} Let +$$f(x) = x^n+c_1x^{n-1}+\cdots + c_{n-1}x + c_n$$ +be a polynomial with complex coefficients $c_i$, where $n>0$. Then there is a +complex number $x$ such that $f(x)=0$. Therefore there are $n$ such complex +numbers (counted with multiplicities). +\end{thm} +\begin{proof} +Using $f(x)/(x-c)$ for a root $c$, we see that the last statement will follow +by induction from the first. We may as well assume that $f(x)\neq 0$ for +$x\in S^1$. This allows us to define $\hat{f}: S^1\rtarr S^1$ by +$\hat{f}(x)=f(x)/|f(x)|$. We proceed to calculate $\text{deg}(\hat{f})$. +Suppose first that $f(x)\neq 0$ for all $x$ such that $|x|\leq 1$. +This allows us to define $h: S^1\times I\rtarr S^1$ by $h(x,t)=f(tx)/|f(tx)|$. +Then $h$ is a homotopy from the constant map at $f(0)/|f(0)|$ to $\hat{f}$, +and we conclude that $\deg(\hat{f})=0$. Suppose next that $f(x)\neq 0$ for all +$x$ such that $|x|\geq 1$. This allows us to define $j:S^1\times I\rtarr S^1$ by +$j(x,t)=k(x,t)/|k(x,t)|$, where +$$k(x,t)=t^nf(x/t)=x^n+t(c_1x^{n-1}+tc_2x^{n-2}+\cdots +t^{n-1}c_n).$$ +Then $j$ is a homotopy from $f_n$ to $\hat{f}$, and we conclude that +$\deg(\hat{f})=n$. One of our suppositions had better be false! +\end{proof} + +It is to be emphasized how technically simple this is, requiring nothing remotely +as deep as complex analysis. Nevertheless, homotopical proofs like this are +relatively recent. Adequate language, elementary as it is, was not developed +until the 1930s. + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Let $p$ be a polynomial function on $\bC$ which has no root on $S^1$. +Show that the number of roots of $p(z) = 0$ with $|z| < 1$ is the degree +of the map $\hat{p}: S^1\rtarr S^1$ specified by $\hat{p}(z) = p(z)/|p(z)|$. +\item Show that any map $f: S^1 \rtarr S^1$ such that $\text{deg}(f)\neq 1$ has +a fixed point. +\item Let $G$ be a topological group and take its identity element $e$ as its basepoint. +Define the pointwise product of loops $\al$ and $\be$ by $(\al\be)(t) = \al (t)\be (t)$. +Prove that $\al\be$ is equivalent to the composition of paths $\be\cdot\al$. +Deduce that $\pi_1(G,e)$ is Abelian. +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{Categorical language and the van Kampen theorem} + +We introduce categorical language and ideas and use them to prove the van Kampen +theorem. This method of computing fundamental groups illustrates the general principle +that calculations in algebraic topology usually work by piecing together a few pivotal +examples by means of general constructions or procedures. + +\section{Categories} + +Algebraic topology concerns mappings from topology to algebra. Category theory gives +us a language to express this. We just record the basic terminology, without being +overly pedantic about it. + +A category\index{category} $\sC$ consists of a collection of objects, a set $\sC(A,B)$ of +morphisms (also called maps) between +any two objects, an identity morphism $\id_A\in\sC(A,A)$ for each object $A$ (usually +abbreviated $\id$), and a composition law +$$\com : \sC (B,C)\times\sC (A,B)\rtarr \sC (A,C)$$ +for each triple of objects $A$, $B$, $C$. Composition must be associative, and identity +morphisms must behave as their names dictate: +$$h\com (g\com f)=(h\com g)\com f,\ \ \ \id\com f =f, \ \tand \ f\com\id=f$$ +whenever the specified composites are defined. A category is ``small''\index{category!small} if +it has a set of objects. + +We have the category $\sS$ of sets and functions, the category $\sU$\index{U@$\sU$} of +topological spaces and continuous functions, the category $\sG$ of groups and homomorphisms, the +category $\sA b$ of Abelian groups and homomorphisms, and so on. + +\section{Functors} + +A functor\index{functor} $F:\sC \rtarr \sD$ is a map of categories. It assigns an object +$F(A)$ of $\sD$ +to each object $A$ of $\sC$ and a morphism $F(f):F(A)\rtarr F(B)$ of $\sD$ to each morphism +$f:A\rtarr B$ of $\sC$ in such a way that +$$F(\id_A)=\id_{F(A)} \ \ \tand \ \ F(g\com f)=F(g)\com F(f).$$ +More precisely, this is a covariant functor\index{functor!covariant}. A contravariant +functor\index{functor!contravariant} $F$ reverses the +direction of arrows, so that $F$ sends $f:A\rtarr B$ to $F(f):F(B)\rtarr F(A)$ and +satisfies $F(g\com f)=F(f)\com F(g)$. A category $\sC$ has an +opposite category\index{category!opposite} +$\sC^{op}$ with the same objects and with $\sC^{op}(A,B)=\sC(B,A)$. A contravariant +functor $F:\sC\rtarr \sD$ is just a covariant functor $\sC^{op}\rtarr \sD$. + +For example, we have forgetful functors from spaces to sets and from Abelian groups +to sets, and we have the free Abelian group functor from sets to Abelian groups. + +\section{Natural transformations} + +A natural transformation\index{natural transformation} +$\al: F\rtarr G$ between functors $\sC\rtarr \sD$ is a map +of functors. It consists of a morphism $\al_A: F(A)\rtarr G(A)$ for each object $A$ of $\sC$ +such that the following diagram commutes for each morphism $f:A\rtarr B$ of $\sC$: +$$\diagram +F(A) \rto^{F(f)} \dto_{\al_A} & F(B) \dto^{\al_B}\\ +G(A) \rto_{G(f)} & G(B).\\ +\enddiagram$$ +Intuitively, the maps $\al_A$ are defined in the same way for every $A$. + +For example, if $F:\sS\rtarr \sA b$ is the functor that sends a set to the free +Abelian group that it generates and $U:\sA b\rtarr \sS$ is the forgetful functor +that sends an Abelian group to its underlying set, then we have a natural inclusion of +sets $S\rtarr UF(S)$. The functors $F$ and $U$ are left adjoint and right +adjoint\index{functor!adjoint}\index{adjoint functors} to each +other, in the sense that we have a natural isomorphism +$$\sA b(F(S),A)\iso \sS(S,U(A))$$ +for a set $S$ and an Abelian group $A$. This just expresses +the ``universal property''\index{universal property} of +free objects: a map of sets $S\rtarr U(A)$ extends uniquely to a homomorphism of groups +$F(S)\rtarr A$. Although we won't bother with a formal definition, the notion of an +adjoint pair of functors will play an important role later on. + +Two categories $\sC$ and $\sD$ are equivalent\index{equivalent!categories} if there are +functors $F:\sC\rtarr \sD$ +and $G:\sD\rtarr \sC$ and natural isomorphisms $FG\rtarr \Id$ and $GF\rtarr \Id$, +where the $\Id$ are the respective identity functors. + +\section{Homotopy categories and homotopy equivalences} + +Let $\sT$\index{T@$\sT$} be the category of spaces $X$ with a chosen basepoint $x\in X$; its +morphisms are continuous maps $X\rtarr Y$ that carry the basepoint of $X$ to the basepoint of $Y$. +The fundamental group specifies a functor $\sT\rtarr\sG$, where $\sG$ is the category of +groups and homomorphisms. + +When we have a (suitable) relation of homotopy between maps in a category $\sC$, we define +the homotopy category\index{category!homotopy}\index{homotopy category} $h\sC$ to be the +category with the same objects as $\sC$ but with +morphisms the homotopy classes of maps. We have the homotopy category $h\sU$ of unbased +spaces. On $\sT$, we require homotopies to map basepoint to basepoint at all times $t$, +and we obtain the homotopy category $h\sT$ of based spaces. The fundamental group is a +homotopy invariant functor\index{functor!homotopy invariant} on $\sT$, in the sense that it +factors through a functor $h\sT\rtarr \sG$. + +A homotopy equivalence\index{homotopy equivalence} in $\sU$ is an isomorphism in $h\sU$. Less +mysteriously, a map +$f:X\rtarr Y$ is a homotopy equivalence if there is a map $g: Y\rtarr X$ such that +both $g\com f\htp \id$ and $f\com g\htp \id$. Working in $\sT$, we obtain the +analogous notion of a based homotopy equivalence. Functors carry isomorphisms to +isomorphisms, so we see that a based homotopy equivalence induces an isomorphism +of fundamental groups. The same is true, less obviously, for unbased homotopy equivalences. + +\begin{prop} +If $f: X\rtarr Y$ is a homotopy equivalence, then +$$f_*:\pi_1(X,x)\rtarr \pi_1(Y,f(x))$$ +is an isomorphism for all $x\in X$. +\end{prop} +\begin{proof} +Let $g:Y\rtarr X$ be a homotopy inverse of $f$. By our homotopy invariance diagram, we +see that the composites +$$\pi_1(X,x)\overto{f_*} \pi_1(Y,f(x)) \overto{g_*} \pi_1(X,(g\com f)(x))$$ +and +$$\pi_1(Y,y)\overto{g_*} \pi_1(X,g(y)) \overto{f_*} \pi_1(Y,(f\com g)(y))$$ +are isomorphisms determined by paths between basepoints given by chosen homotopies +$g\com f\htp\id$ and $f\com g\htp\id$. Therefore, in each displayed composite, the +first map is a monomorphism and the second is an epimorphism. Taking $y=f(x)$ in +the second composite, we see that the second map in the first composite is an +isomorphism. Therefore so is the first map. +\end{proof} + +A space $X$ is said to be contractible\index{contractible space} if it is homotopy +equivalent to a point. + +\begin{cor} +The fundamental group of a contractible space is zero. +\end{cor} + +\section{The fundamental groupoid} + +While algebraic topologists often concentrate on connected spaces with chosen basepoints, +it is valuable to have a way of studying fundamental groups that does not require such +choices. For this purpose, we define the ``fundamental groupoid''\index{fundamental groupoid} +$\PI(X)$ of a space $X$ to be the category whose objects are the points of $X$ and whose +morphisms $x\rtarr y$ +are the equivalence classes of paths from $x$ to $y$. Thus the set of endomorphisms of +the object $x$ is exactly the fundamental group $\pi_1(X,x)$. + +The term ``groupoid''\index{groupoid} is used for a category all morphisms of which are +isomorphisms. +The idea is that a group may be viewed as a groupoid with a single object. Taking morphisms +to be functors, we obtain the category $\sG\sP$ of groupoids. Then we may view $\PI$ as a +functor $\sU\rtarr \sG\sP$. + +There is a useful notion of a skeleton\index{skeleton!of a category} $sk\sC$ of a category $\sC$. +This is a ``full'' subcategory \index{full subcategory} with one object from each isomorphism +class of objects of $\sC$, ``full'' meaning +that the morphisms between two objects of $sk\sC$ are all of the morphisms between these +objects in $\sC$. The inclusion functor $J: sk\sC\rtarr \sC$ is an +equivalence of categories. An inverse functor $F:\sC\rtarr sk\sC$ is obtained by letting +$F(A)$ be the unique object in $sk\sC$ that is isomorphic to $A$, choosing an isomorphism +$\al_A: A\rtarr F(A)$, and defining $F(f)=\al_B\com f\com\al_A^{-1}:F(A)\rtarr F(B)$ for +a morphism $f:A\rtarr B$ in $\sC$. We choose $\al$ to be the identity morphism if $A$ is +in $sk\sC$, and then $FJ=\Id$; the $\al_A$ specify a natural isomorphism $\al:\Id\rtarr JF$. + +A category $\sC$ is said to be connected\index{category!connected} if any two of its objects +can be connected by a sequence of morphisms. For example, a sequence $A \longleftarrow B \rtarr C$ +connects $A$ to $C$, although there need be no morphism $A\rtarr C$. However, a groupoid $\sC$ is +connected if and only if any two of its objects are isomorphic. The group of endomorphisms of +any object $C$ is then a skeleton of $\sC$. Therefore the previous paragraph specializes to give +the following relationship between the fundamental group and the fundamental groupoid of a +path connected space $X$. + +\begin{prop} Let $X$ be a path connected space. For each point $x\in X$, the inclusion +$\pi_1(X,x)\rtarr \PI(X)$ is an equivalence of categories. +\end{prop} +\begin{proof} +We are regarding $\pi_1(X,x)$ as a category with a single object $x$, and it is a +skeleton of $\PI(X)$. +\end{proof} + +\section{Limits and colimits} + +Let $\sD$ be a small category and let $\sC$ be any category. A $\sD$-shaped diagram +in $\sC$ is a functor $F:\sD\rtarr \sC$. A morphism $F\rtarr F'$ of $\sD$-shaped +diagrams\index{diagram!$\sD$-shaped} is a natural transformation, and we have the +category $\sD[\sC]$ of $\sD$-shaped +diagrams in $\sC$. Any object $C$ of $\sC$ determines the constant diagram $\ul{C}$ that +sends each object of $\sD$ to $C$ and sends each morphism of $\sD$ to the identity +morphism of $C$. + +The colimit, $\colim F$,\index{colimit} of a $\sD$-shaped diagram $F$ is an object of $\sC$ +together with a morphism of diagrams $\io: F\rtarr \ul{\colim F}$ that is +initial among all such morphisms. This means that if $\et: F\rtarr \ul A$ is +a morphism of diagrams, then there is a unique map $\tilde{\et}: \colim F\rtarr A$ +in $\sC$ such that $\tilde{\eta}\com \io=\et$. Diagrammatically, this property +is expressed by the assertion that, for each map $d:D\rtarr D'$ in $\sD$, we have a +commutative diagram +$$\diagram +F(D)\rrto^{F(d)} \drto_{\io} \ddrto_{\et} && F(D') \dlto^{\io} \ddlto^{\et}\\ + & \colim F \dto^{\tilde{\et}} & \\ + & A. &\\ +\enddiagram$$ + +The limit\index{limit} of $F$ is defined by reversing arrows: it is an object $\lim F$ of $\sC$ +together with a morphism of diagrams $\pi: \ul{\lim F}\rtarr F $ that is +terminal among all such morphisms. This means that if $\epz: \ul A\rtarr F$ is +a morphism of diagrams, then there is a unique map $\tilde{\epz}: A\rtarr\lim F$ +in $\sC$ such that $\pi\com\tilde{\epz}=\epz$. Diagrammatically, this property +is expressed by the assertion that, for each map $d:D\rtarr D'$ in $\sD$, we have a +commutative diagram +$$\diagram +F(D)\rrto^{F(d)} && F(D') \\ + & \lim F \ulto^{\pi} \urto_{\pi} & \\ + & A. \uto_{\tilde{\epz}} \uulto^{\epz} \uurto_{\epz} &\\ +\enddiagram$$ + +If $\sD$ is a set regarded as a discrete category\index{category!discrete} (only identity +morphisms), then +colimits and limits indexed on $\sD$ are coproducts\index{coproduct} and products\index{product} +indexed on the set $\sD$. +Coproducts are disjoint unions in $\sS$ or $\sU$, wedges (or one-point unions)\index{wedge} in +$\sT$, free products in $\sG$, and direct sums in $\sA b$. Products are Cartesian products in +all of these categories; more precisely, they are Cartesian products of underlying sets, +with additional structure. If $\sD$ is the category displayed schematically as +$$\diagram +e & d \lto \rto & f & \text{or} & d \rto<.5ex> \rto<-.5ex> & d',\\ +\enddiagram$$ +where we have displayed all objects and all non-identity morphisms, then the co\-limits +indexed on $\sD$ are called pushouts\index{pushout} or coequalizers\index{coequalizer}, +respectively. Similarly, if $\sD$ is +displayed schematically as +$$\diagram +e \rto & d & f \lto & \text{or} & d \rto<.5ex> \rto<-.5ex> & d',\\ +\enddiagram$$ +then the limits indexed on $\sD$ are called pullbacks\index{pullback} or +equalizers,\index{equalizer} respectively. + +A given category may or may not have all colimits, and it may have some but not others. +A category is said to be cocomplete\index{category!cocomplete} if it has all colimits, +complete\index{category!complete} if it has all +limits. The categories $\sS$, $\sU$, $\sT$, $\sG$, and $\sA b$ are complete and cocomplete. +If a category has coproducts and coequalizers, then it is cocomplete, and similarly for +completeness. The proof is a worthwhile exercise. + +\section{The van Kampen theorem} + +The following is a modern dress treatment of the van Kampen theorem. I should admit that, +in lecture, it may make more sense not to introduce the fundamental groupoid and to +go directly to the fundamental group statement. The direct proof is shorter, but +not as conceptual. However, as far as I know, the deduction of the fundamental group version +of the van Kampen theorem from the fundamental groupoid version does not appear in the literature +in full generality. The proof well illustrates how to manipulate colimits formally. We have used +the van Kampen theorem as an excuse to introduce some basic categorical language, and we shall use +that language heavily in our treatment of covering spaces in the next chapter. + +\begin{thm}[van Kampen]\index{van Kampen theorem|(} +Let $\sO=\sset{U}$ be a cover of a space $X$ by path connected open subsets such that +the intersection of finitely many subsets in $\sO$ is again in $\sO$. Regard $\sO$ as +a category whose morphisms are the inclusions of subsets and observe that the functor +$\PI$, restricted to the spaces and maps in $\sO$, gives a diagram +$$\PI|\sO: \sO\rtarr \sG\sP$$ +of groupoids. The groupoid $\PI(X)$ is the colimit of this diagram. In symbols, +$$\PI (X) \iso \colim_{U\in\sO} \PI(U).$$ +\end{thm} +\begin{proof} +We must verify the universal property. For a groupoid $\sC$ and a map +$\et: \PI|\sO\rtarr \ul{\sC}$ of $\sO$-shaped diagrams of groupoids, we must +construct a map $\tilde{\et}: \PI(X)\rtarr \sC$ of groupoids that restricts to $\et_U$ on +$\PI(U)$ for each $U\in\sO$. On objects, that is on points of $X$, we must define +$\tilde{\et}(x)=\et_U(x)$ for $x\in U$. This is independent of the choice of $U$ +since $\sO$ is closed under finite intersections. If a path $f:x\to y$ lies entirely +in a particular $U$, then we must define $\tilde{\et}[f]=\et([f])$. Again, since $\sO$ +is closed under finite intersections, this specification is independent of the choice +of $U$ if $f$ lies entirely in more than one $U$. Any path $f$ is the composite of +finitely many paths $f_i$, each of which does lie in a single $U$, and we must +define $\tilde{\et}[f]$ to be the composite of the $\tilde{\et}[f_i]$. Clearly this +specification will give the required unique map $\tilde{\et}$, provided that $\tilde{\et}$ +so specified is in fact well defined. Thus suppose that $f$ is equivalent to $g$. The +equivalence is given by a homotopy $h:f\htp g$ through paths $x\to y$. We may subdivide +the square $I\times I$ into subsquares, each of which is mapped into one of the $U$. +We may choose the subdivision so that the resulting subdivision of $I\times\sset{0}$ refines the +subdivision used to decompose $f$ as the composite of paths $f_i$, and similarly for $g$ and +the resulting subdivision of $I\times\sset{1}$. We see that the relation $[f]=[g]$ in $\PI(X)$ is +a consequence of a finite number of relations, each of which holds in one of the $\PI(U)$. +Therefore $\tilde\et([f])=\tilde\eta([g])$. This verifies the universal property and proves +the theorem. +\end{proof} + +The fundamental group version of the van Kampen theorem ``follows formally.'' That +is, it is an essentially categorical consequence of the version just proved. Arguments +like this are sometimes called proof by categorical nonsense. + +\begin{thm}[van Kampen]\index{van Kampen theorem|)} +Let $X$ be path connected and choose a basepoint $x\in X$. Let $\sO$ +be a cover of $X$ by path connected open subsets such that the intersection of finitely +many subsets in $\sO$ is again in $\sO$ and $x$ is in each $U\in\sO$. Regard $\sO$ as +a category whose morphisms are the inclusions of subsets and observe that the functor +$\pi_1(-,x)$, restricted to the spaces and maps in $\sO$, gives a diagram +$$\pi_1|\sO: \sO\rtarr \sG$$ +of groups. The group $\pi_1(X,x)$ is the colimit of this diagram. In symbols, +$$\pi_1(X,x) \iso \colim_{U\in\sO} \pi_1(U,x).$$ +\end{thm} + +We proceed in two steps. + +\begin{lem} The van Kampen theorem holds when the cover $\sO$ is finite. +\end{lem} +\begin{proof} This step is based on the nonsense above about skeleta of categories. +We must verify the universal property, this time in the category of groups. For a group $G$ +and a map $\et: \pi_1|\sO\rtarr \ul{G}$ of $\sO$-shaped diagrams of groups, we must show +that there is a unique homomorphism $\tilde{\et}: \pi_1(X,x)\rtarr G$ that restricts to +$\et_U$ on $\pi_1(U,x)$. Remember that we think of a group as a groupoid with a single +object and with the elements of the group as the morphisms. The inclusion of categories +$J: \pi_1(X,x)\rtarr \PI(X)$ is an equivalence. An inverse equivalence $F: \PI(X)\rtarr \pi_1(X,x)$ +is determined by a choice of path classes $x\rtarr y$ for $y\in X$; we choose $c_x$ when +$y=x$ and so ensure that $F\com J = \Id$. Because the cover $\sO$ is finite and closed +under finite intersections, we can choose our paths inductively so that the path $x\rtarr y$ +lies entirely in $U$ whenever $y$ is in $U$. This ensures that the chosen paths determine +compatible inverse equivalences +$F_U: \PI(U)\rtarr \pi_1(U,x)$ to the inclusions $J_U: \pi_1(U,x)\rtarr \PI(U)$. Thus the functors +$$\diagram +\PI(U)\rto^{F_U} & \pi_1(U,x) \rto^(0.6){\et_U} & G +\enddiagram$$ +specify an $\sO$-shaped diagram of groupoids $\PI|\sO\rtarr \ul{G}$. By the fundamental groupoid +version of the van Kampen theorem, there is a unique map of groupoids +$$\xi: \PI(X)\rtarr G$$ +that restricts to $\et_U\com F_U$ on $\PI(U)$ for each $U$. The composite +$$\diagram +\pi_1(X,x)\rto^{J} & \PI(X)\rto^(0.6){\xi} & G +\enddiagram$$ +is the required homomorphism $\tilde{\et}$. It restricts to $\et_U$ on $\pi_1(U,x)$ +by a little ``diagram chase'' and the fact that $F_U\com J_U=\Id$. It is unique because +$\xi$ is unique. In fact, if we are given $\tilde{\et}: \pi_1(X,x)\rtarr G$ that restricts +to $\et_U$ on each $\pi_1(U,x)$, then $\tilde{\et}\com F:\PI(X)\rtarr G$ restricts to +$\et_U\com F_U$ on each $\PI(U)$; therefore $\xi = \tilde{\et}\com F$ and thus +$\xi\com J = \tilde\et$. +\end{proof} + +\begin{proof}[Proof of the van Kampen theorem] +We deduce the general case from the case just proved. Let $\sF$ be the set of those finite +subsets of the cover $\sO$ that are closed under finite intersection. For $\sS\in \sF$, +let $U_{\sS}$ be the union of the $U$ in $\sS$. Then $\sS$ is a cover of $U_{\sS}$ to +which the lemma applies. Thus +$$ \colim_{U\in\sS}\pi_1(U,x)\iso \pi_1(U_{\sS},x).$$ +Regard $\sF$ as a category with a morphism $\sS\rtarr \sT$ whenever $U_{\sS}\subset U_{\sT}$. +We claim first that +$$\colim_{\sS\in\sF}\pi_1(U_{\sS},x)\iso \pi_1(X,x).$$ +In fact, by the usual subdivision argument, any loop $I\rtarr X$ and any +equivalence $h: I\times I\rtarr X$ between loops has image in some $U_{\sS}$. +This implies directly that $\pi_1(X,x)$, together with the homomorphisms +$\pi_1(U_{\sS},x)\rtarr \pi_1(X,x)$, has the universal property that characterizes +the claimed colimit. We claim next that +$$\colim_{U\in\sO}\pi_1(U,x)\iso \colim_{\sS\in\sF}\pi_1(U_{\sS},x),$$ +and this will complete the proof. Substituting in the colimit on the right, we have +$$\colim_{\sS\in\sF}\pi_1(U_{\sS},x)\iso \colim_{\sS\in\sF}\colim_{U\in\sS}\pi_1(U,x).$$ +By a comparison of universal properties, this iterated colimit is isomorphic to the single +colimit +$$\colim_{(U,\sS)\in (\sO,\sF)}\pi_1(U,x).$$ +Here the indexing category $(\sO,\sF)$ has objects the pairs $(U,\sS)$ with $U\in \sS$; there +is a morphism $(U,\sS)\rtarr (V,\sT)$ whenever both $U\subset V$ and $U_{\sS}\subset U_{\sT}$. +A moment's reflection on the relevant universal properties should convince the reader of the +claimed identification of colimits: the system on the right differs from the system on the left +only in that the homomorphisms $\pi_1(U,x)\rtarr \pi_1(V,x)$ occur many times in the system +on the right, each appearance making the same contribution to the colimit. If we assume known +a priori that colimits of groups exist, we can formalize this as follows. We have a functor +$\sO\rtarr \sF$ that sends $U$ to the singleton set $\sset{U}$ and thus a functor +$\sO\rtarr (\sO,\sF)$ that sends $U$ to $(U,\sset{U})$. The functor $\pi_1(-,x):\sO\rtarr \sG$ +factors through $(\sO,\sF)$, hence we have an induced map of colimits +$$\colim_{U\in\sO}\pi_1(U,x)\rtarr \colim_{(U,\sS)\in (\sO,\sF)}\pi_1(U,x).$$ +Projection to the first coordinate gives a functor $(\sO,\sF)\rtarr \sO$. Its composite with +$\pi_1(-,x):\sO\rtarr \sG$ defines the colimit on the right, hence we have an induced map of +colimits +$$\colim_{(U,\sS)\in (\sO,\sF)}\pi_1(U,x)\rtarr \colim_{U\in\sO}\pi_1(U,x).$$ +These maps are inverse isomorphisms. +\end{proof} + +\section{Examples of the van Kampen theorem} + +So far, we have only computed the fundamental groups of the circle and of contractible +spaces. The van Kampen theorem lets us extend these calculations. We now drop notation +for the basepoint, writing $\pi_1(X)$ instead of $\pi_1(X,x)$. + +\begin{prop} +Let $X$ be the wedge of a set of path connected based spaces $X_i$, each of which +contains a contractible neighborhood $V_i$ of its basepoint. Then $\pi_1(X)$ is the +coproduct (= free product)\index{free product} of the groups $\pi_1(X_i)$. +\end{prop} +\begin{proof} +Let $U_i$ be the union of $X_i$ and the $V_j$ for $j\neq i$. We apply the van Kampen +theorem with $\sO$ taken to be the $U_i$ and their finite intersections. Since any +intersection of two or more of the $U_i$ is contractible, the intersections make no +contribution to the colimit and the conclusion follows. +\end{proof} + +\begin{cor} +The fundamental group of a wedge of circles is a free group with one generator for +each circle. +\end{cor} + +Any compact surface is homeomorphic to a sphere, or to a connected sum of tori\index{torus} +$T^2=S^1\times S^1$, or to a connected sum of projective planes\index{projective plane} +$\bR P^2=S^2/\bZ_2$ (where we write $\bZ_2 = \bZ/2\bZ$). +We shall see shortly that $\pi_1(\bR P^2) =\bZ_2$. We also have the following observation, +which is immediate from the universal property of products. Using this information, +it is an exercise to compute the fundamental group of any compact surface from the +van Kampen theorem. + +\begin{lem} +For based spaces $X$ and $Y$, $\pi_1(X\times Y)\iso \pi_1(X)\times \pi_1(Y)$. +\end{lem} + +We shall later use the following application of the van Kampen theorem to prove +that any group is the fundamental group of some space. We need a definition. + +\begin{defn} +A space $X$ is said to be simply connected if it is path connected and satisfies +$\pi_1(X)=0$. +\end{defn} + +\begin{prop} +Let $X=U\cup V$, where $U$, $V$, and $U\cap V$ are path connected open neighborhoods of +the basepoint of $X$ and $V$ is simply connected. Then $\pi_1(U)\rtarr \pi_1(X)$ is +an epimorphism whose kernel is the smallest normal subgroup of $\pi_1(U)$ that contains +the image of $\pi_1(U\cap V)$. +\end{prop} +\begin{proof} +Let $N$ be the cited kernel and consider the diagram +$$\diagram + & \pi_1(U) \drto \drrto & & \\ +\pi_1(U\cap V) \drto \urto & &\pi_1(X) \rdashed^(0.4){\xi}|>\tip & \pi_1(U)/N\\ +& \pi_1(V)=0 \urto \urrto & & \\ +\enddiagram$$ +The universal property gives rise to the map $\xi$, and +$\xi$ is an isomorphism since, by an easy algebraic inspection, $\pi_1(U)/N$ is the +pushout in the category of groups of the homomorphisms $\pi_1(U\cap V)\rtarr \pi_1(U)$ +and $\pi_1(U\cap V)\rtarr 0$. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Compute the fundamental group of the two-holed torus (the compact surface of genus $2$ +obtained by sewing together two tori along the boundaries of an open disk removed from each). +\item The Klein bottle\index{Klein bottle} +$K$ is the quotient space of $S^1\times I$ obtained by identifying +$(z,0)$ with $(z^{-1},1)$ for $z\in S^1$. Compute $\pi_1(K)$. +\item$*$ Let $X = \{ (p,q)| p \neq -q\}\subset S^n\times S^n$. Define a map $f: S^n\rtarr X$ by +$f(p) = (p,p)$. Prove that $f$ is a homotopy equivalence. +\item Let $\sC$ be a category that has all coproducts and coequalizers. Prove that $\sC$ is +cocomplete (has all colimits). Deduce formally, by use of opposite categories, that a category +that has all products and equalizers is complete. +\end{enumerate} + +\chapter{Covering spaces} + +We run through the theory of covering spaces and their relationship to +fundamental groups and fundamental groupoids. This is standard material, +some of the oldest in algebraic topology. However, I know of no published +source for the use that we shall make of the orbit category $\sO(\pi_1(B,b))$ +in the classification of coverings of a space $B$. This point of view gives us +the opportunity to introduce some ideas that are central to equivariant +algebraic topology, the study of spaces with group actions. In any case, this +material is far too important to all branches of mathematics to omit. + +\section{The definition of covering spaces} + +While the reader is free to think about locally contractible spaces, +weaker conditions are appropriate for the full generality of the theory +of covering spaces. A space $X$ is +said to be locally path connected\index{locally path connected space} if for +any $x\in X$ and any neighborhood +$U$ of $x$, there is a smaller neighborhood V of $x$ each of whose points +can be connected to $x$ by a path in $U$. This is equivalent to the +seemingly more stringent requirement that the topology of $X$ have a basis +consisting of path connected open sets. In fact, if $X$ is locally path +connected and $U$ is an open neighborhood of a point $x$, then the set +$$V=\{y\,|\,y \ \text{can be connected to}\ x \ \text{by a path in}\ U\}$$ +is a path connected open neighborhood of $x$ that is contained in $U$. +Observe that if $X$ is connected and locally path connected, then it is +path connected. Throughout this chapter, we assume that all given spaces +are connected and locally path connected. + +\begin{defn} +A map $p: E\rtarr B$ is a covering (or cover, or covering +space)\index{cover}\index{covering}\index{covering space} if it is +surjective and if each +point $b\in B$ has an open neighborhood $V$ such that each component of +$p^{-1}(V)$ is open in $E$ and is mapped homeomorphically onto $V$ by $p$. +We say that a path connected open subset $V$ with this property is a +fundamental neighborhood\index{fundamental neighborhood} of $B$. We call +$E$ the total space,\index{total space} $B$ the base +space,\index{base space} and $F_b=p^{-1}(b)$ a fiber\index{fiber} of the +covering $p$. +\end{defn} + +Any homeomorphism is a cover. A product of covers is a cover. The projection +$\bR\rtarr S^1$ is a cover. Each $f_n: S^1\rtarr S^1$ is a cover. The +projection $S^n\rtarr \bR P^n$ is a cover, where the real projective +space $\bR P^n$ is obtained from $S^n$ by identifying antipodal points. If +$f: A\rtarr B$ is a map (where $A$ is connected and locally path connected) +and $D$ is a component of the pullback of $f$ along $p$, then $p: D\rtarr A$ +is a cover. + +\section{The unique path lifting property} + +The following result is abstracted from what we saw in the case of the +particular cover $\bR\rtarr S^1$. It describes the behavior of $p$ with +respect to path classes and fundamental groups. + +\begin{thm}[Unique path lifting]\index{unique path lifting theorem} Let +$p: E\rtarr B$ be a covering, let $b\in B$, +and let $e,e'\in F_b$. +\begin{enumerate} +\item[(i)] A path $f:I\rtarr B$ with $f(0)=b$ lifts uniquely to a path +$g:I\rtarr E$ such that $g(0)=e$ and $p\com g=f$. +\item[(ii)] Equivalent paths $f\htp f':I\rtarr B$ that start at $b$ lift +to equivalent paths $g\htp g': I\rtarr E$ that start at $e$, hence +$g(1)=g'(1)$. +\item[(iii)] $p_*: \pi_1(E,e)\rtarr \pi_1(B,b)$ is a monomorphism. +\item[(iv)] $p_*(\pi_1(E,e'))$ is conjugate to $p_*(\pi_1(E,e))$. +\item[(v)] As $e'$ runs through $F_b$, the groups $p_*(\pi_1(E,e'))$ +run through all conjugates of $p_*(\pi_1(E,e))$ in $\pi_1(B,b)$. +\end{enumerate} +\end{thm} +\begin{proof} For (i), subdivide $I$ into subintervals each of which maps to a +fundamental neighborhood under $f$, and lift $f$ to $g$ inductively by use of +the prescribed homeomorphism property of fundamental neighborhoods. For (ii), +let $h:I\times I\rtarr B$ be a homotopy $f\htp f'$ through paths +$b\rtarr b'$. Subdivide the square into subsquares each of which maps to a +fundamental neighborhood under $f$. Proceeding inductively, we see that $h$ lifts +uniquely to a homotopy $H:I\times I\rtarr E$ such that $H(0,0)=e$ and $p\com H=h$. +By uniqueness, $H$ is a homotopy $g\htp g'$ through paths $e\rtarr e'$, where +$g(1)=e'=g'(1)$. Parts (iii)--(v) are formal consequences of (i) and (ii), as we +shall see in the next section. +\end{proof} + +\begin{defn} +A covering $p: E\rtarr B$ is regular\index{covering!regular} if $p_*(\pi_1(E,e))$ is a +normal subgroup of $\pi_1(B,b)$. It is universal\index{covering!universal} if $E$ is +simply connected. +\end{defn} + +As we shall explain in \S4, for a universal cover $p: E\rtarr B$, the +elements of $F_b$ are in bijective correspondence with the elements of $\pi_1(B,b)$. +We illustrate the force of this statement. + +\begin{exmp} +For $n\geq 2$, $S^n$ is a universal cover of $\bR P^n$. Therefore $\pi_1(\bR P^n)$ +has only two elements. There is a unique group with two elements, and this proves +our earlier claim that $\pi_1(\bR P^n)=\bZ_2$. +\end{exmp} + +\section{Coverings of groupoids} + +Much of the theory of covering spaces can be recast conceptually in terms +of fundamental groupoids. This point of view separates the essentials +of the topology from the formalities and gives a convenient language in +which to describe the algebraic classification of coverings. + +\begin{defn} (i) Let $\sC$ be a category and $x$ be an object of $\sC$. The +category $x\backslash \sC$ of objects under $x$ has objects the maps +$f: x\rtarr y$ in $\sC$; for objects $f:x\rtarr y$ and $g: x\rtarr z$, the +morphisms $\ga: f\rtarr g$ in $x\backslash\sC$ are the morphisms +$\ga: y\rtarr z$ in $\sC$ such that $\ga\com f = g:x\rtarr z$. Composition and +identity maps are given by composition and identity maps in $\sC$. When +$\sC$ is a groupoid, $\ga = g\com f^{-1}$, and the objects of +$x\backslash \sC$ therefore determine the category. + +(ii) Let $\sC$ be a small groupoid. Define the star\index{star} of $x$, denoted $St(x)$ +or $St_{\sC}(x)$, to be the set of objects of $x\backslash \sC$, that is, the +set of morphisms of $\sC$ with source $x$. Write $\sC(x,x)=\pi(\sC,x)$ for the +group of automorphisms of the object $x$. + +(iii) Let $\sE$ and $\sB$ be small connected groupoids. A +covering\index{covering!of a groupoid} +$p: \sE \rtarr \sB$ is a functor that is surjective on objects and +restricts to a bijection +$$p: St(e)\rtarr St(p(e))$$ +for each object $e$ of $\sE$. For an object $b$ of $\sB$, let $F_b$ +denote the set of objects of $\sE$ such that $p(e) = b$. Then +$p^{-1}(St(b))$ is the disjoint union over $e\in F_b$ of $St(e)$. +\end{defn} + +Parts (i) and (ii) of the unique path lifting theorem can be restated +as follows. + +\begin{prop} If $p: E\rtarr B$ is a covering of spaces, then the induced +functor $\PI(p): \PI(E)\rtarr \PI(B)$ is a covering of groupoids. +\end{prop} + +Parts (iii), (iv), and (v) of the unique path lifting theorem are categorical +consequences that apply to any covering of groupoids, where they read as follows. + +\begin{prop} Let $p: \sE\rtarr \sB$ be a covering of groupoids, let $b$ +be an object of $\sB$, and let $e$ and $e'$ be objects of $F_b$. +\begin{enumerate} +\item[(i)] $p: \pi(\sE ,e)\rtarr \pi(\sB ,b)$ is a monomorphism. +\item[(ii)] $p(\pi(\sE ,e'))$ is conjugate to $p(\pi(\sE,e))$. +\item[(iii)] As $e'$ runs through $F_b$, the groups $p(\pi(E,e'))$ +run through all conjugates of $p(\pi(\sE ,e))$ in $\pi(\sB ,b)$. +\end{enumerate} +\end{prop} +\begin{proof} For (i), if $g,g'\in \pi(\sE ,e)$ and $p(g) = p(g')$, then +$g = g'$ by the injectivity of $p$ on $St(e)$. For (ii), there is a map +$g: e\rtarr e'$ since $\sE$ is connected. Conjugation by $g$ gives a +homomorphism $\pi(\sE ,e)\rtarr \pi(\sE, e')$ that maps under $p$ to +conjugation of $\pi(\sB,b)$ by its element $p(g)$. +For (iii), the surjectivity of $p$ on $St(e)$ gives that any $f\in\pi(\sB ,b)$ +is of the form $p(g)$ for some $g\in St(e)$. If $e'$ is the target of $g$, +then $p(\pi(\sE,e'))$ is the conjugate of $p(\pi(\sE,e))$ by $f$. +\end{proof} + +The fibers $F_b$ of a covering of groupoids are related by translation functions. + +\begin{defn} Let $p: \sE\rtarr \sB$ be a covering of groupoids. Define the +fiber translation functor\index{fiber translation functor} $T=T(p): \sB\rtarr \sS$ +as follows. For an object $b$ of +$\sB$, $T(b) = F_b$. For a morphism $f: b\rtarr b'$ of $\sB$, $T(f): F_b\rtarr F_{b'}$ +is specified by $T(f)(e) = e'$, where $e'$ is the target of the unique $g$ in +$St(e)$ such that $p(g)=f$. +\end{defn} + +It is an exercise from the definition of a covering of a groupoid to verify that +$T$ is a well defined functor. For a covering space $p: E\rtarr B$ and a path +$f: b\rtarr b'$, $T(f): F_b\rtarr F_{b'}$ is given by $T(f)(e)=g(1)$ where $g$ +is the path in $E$ that starts at $e$ and covers $f$. + +\begin{prop} Any two fibers $F_b$ and $F_{b'}$ of a covering of groupoids have the +same cardinality. Therefore any two fibers of a covering of spaces have the same +cardinality. +\end{prop} +\begin{proof} +For $f: b\rtarr b'$, $T(f): F_b\rtarr F_{b'}$ is a bijection with inverse $T(f^{-1})$. +\end{proof} + +\section{Group actions and orbit categories} + +The classification of coverings is best expressed in categorical language +that involves actions of groups and groupoids on sets. + +A (left) action of a group $G$\index{group action} on a set $S$ is a function +$G\times S\rtarr S$ such that $es=s$ (where $e$ is the identity element) and +$(g'g)s=g'(gs)$ for all $s\in S$. The {\em isotropy group}\index{isotropy group} +$G_s$ of a point $s$ is the subgroup $\{ g| gs=s\}$ of $G$. +An action is {\em free}\index{group action!free} if $gs=s$ implies $g=e$, that is, +if $G_s=e$ for every $s\in S$. + +The orbit generated by a point $s$ is $\{ gs|g\in G\}$. An action is +{\em transitive}\index{group action!transitive} +if for every pair $s,s'$ of elements of $S$, there is an element $g$ of $G$ such +that $gs=s'$. Equivalently, $S$ consists of a single orbit. If $H$ is a subgroup +of $G$, the set $G/H$ of cosets $gH$ is a transitive $G$-set. When $G$ acts +transitively on a set $S$, we obtain an isomorphism of $G$-sets between $S$ and +the $G$-set $G/G_s$ for any fixed $s\in S$ by sending $gs$ to the coset $gG_s$. + +The following lemma describes the group of automorphisms of a transitive \linebreak +$G$-set $S$. +For a subgroup $H$ of $G$, let $NH$ denote the normalizer of $H$ in $G$ and define +$WH=NH/H$. Such quotient groups $WH$ are sometimes called Weyl groups.\index{Weyl group} + +\begin{lem} Let $G$ act transitively on a set $S$, choose $s\in S$, and let $H=G_s$. +Then $WH$ is isomorphic to the group {\em Aut}$_G(S)$\index{AutGS@Aut$_G(S)$} of automorphisms +of the $G$-set $S$. +\end{lem} +\begin{proof} +For $n\in NH$ with image $\bar{n}\in WH$, define an automorphism $\ph(\bar{n})$ of $S$ +by $\ph(\bar{n})(gs)=gns$. For an automorphism $\ph$ of $S$, we have $\ph(s)=ns$ for some +$n\in G$. For $h\in H$, $hns=\ph(hs)=\ph(s)=ns$, hence $n^{-1}hn\in G_s=H$ and $n\in NH$. +Clearly $\ph=\ph(\bar{n})$, and it is easy to check that this bijection between $WH$ and +Aut$_G(S)$ is an isomorphism of groups. +\end{proof} + +We shall also need to consider $G$-maps between different $G$-sets $G/H$. + +\begin{lem} A $G$-map $\al: G/H\rtarr G/K$ has the form $\al(gH) = g\ga K$, where +the element $\ga\in G$ satisfies $\ga^{-1} h \ga\in K$ for all $h\in H$. +\end{lem} +\begin{proof} +If $\al(eH) = \ga K$, then the relation +$$\ga K = \al (eH) = \al(hH) = h\al(eH) = h\ga K$$ +implies that $\ga^{-1}h\ga\in K$ for $h\in H$. +\end{proof} + +\begin{defn} The category $\sO(G)$ of canonical +orbits\index{category!of canonical orbits}\index{orbit category} has +objects the $G$-sets $G/H$ and morphisms the $G$-maps of $G$-sets. +\end{defn} + +The previous lemmas give some feeling for the structure of $\sO(G)$ and lead +to the following alternative description. + +\begin{lem} The category $\sO(G)$ is isomorphic to the category $\sG$ whose +objects are the subgroups of $G$ and whose morphisms are the distinct +subconjugacy relations $\ga^{-1}H\ga \subset K$ for $\ga\in G$. +\end{lem} + +If we regard $G$ as a category with a single object, then a (left) action of +$G$ on a set $S$ is the same thing as a covariant functor $G\rtarr \sS$. +(A right action is the same thing as a contravariant functor.) If $\sB$ is +a small groupoid, it is therefore natural to think of a covariant functor +$T:\sB\rtarr \sS$ as a generalization of a group action.\index{groupoid action} For +each object $b$ of $\sB$, $T$ restricts to an action of $\pi(\sB ,b)$ on $T(b)$. We +say that the functor $T$ is {\em transitive}\index{groupoid action!transitive} if +this group action is transitive for each object $b$. If $\sB$ is connected, this +holds for all objects $b$ if it holds for any one object $b$. + +For example, for a covering of groupoids $p: \sE\rtarr \sB$, the fiber translation +functor $T$ restricts to give an action of $\pi(\sB,b)$ on the set $F_b$. For $e\in F_b$, +the isotropy group of $e$ is precisely $p(\pi(\sE,e))$. That is, $T(f)(e)=e$ if and only +if the lift of $f$ to an element of $St(e)$ is an automorphism of $e$. Moreover, the +action is transitive since there is an isomorphism in $\sE$ connecting any two points of +$F_b$. Therefore, as a $\pi(\sB,b)$-set, +$$F_b\iso \pi(\sB,b)/p(\pi(\sE,e)).$$ + +\begin{defn} A covering $p: \sE\rtarr \sB$ of groupoids is +regular\index{covering!regular} if +$p(\pi(\sE,e))$ is a normal subgroup of $\pi(\sB,b)$. It is +universal\index{covering!universal} if $p(\pi(\sE,e))=\sset{e}$. +Clearly a covering space is regular or universal if and only if its +associated covering of fundamental groupoids is regular or universal. +\end{defn} + +A covering of groupoids is universal if and only if $\pi(\sB,b)$ acts freely on $F_b$, +and then $F_b$ is isomorphic to $\pi(\sB,b)$ as a $\pi(\sB,b)$-set. Specializing +to covering spaces, this sharpens our earlier claim that the elements of $F_b$ and +$\pi_1(B,b)$ are in bijective correspondence. + +\section{The classification of coverings of groupoids} + +Fix a small connected groupoid $\sB$ throughout this section and the next. We explain +the classification of coverings of $\sB$. This gives an algebraic prototype for +the classification of coverings of spaces. We begin with a result that should be +called the fundamental theorem\index{fundamental theorem!of covering groupoid theory} +of covering groupoid theory. We assume once and for +all that all given groupoids are small and connected. + +\begin{thm} +Let $p: \sE\rtarr \sB$ be a covering of groupoids, let $\sX$ be a groupoid, and let +$f: \sX\rtarr \sB$ be a functor. Choose a base object $x_0\in \sX$, let $b_0=f(x_0)$, and +choose $e_0\in F_{b_0}$. Then there exists a functor $g: \sX\rtarr \sE$ such that +$g(x_0)=e_0$ and $p\com g= f$ if and only if +$$f(\pi(\sX,x_0))\subset p(\pi(\sE,e_0))$$ +in $\pi(\sB,b_0)$. When this condition holds, there is a unique such functor $g$. +\end{thm} +\begin{proof} +If $g$ exists, its properties directly imply that $\im(f)\subset\im(p)$. For an +object $x$ of $\sX$ and a map $\al: x_0\rtarr x$ in $\sX$, let $\tilde \al$ +be the unique element of $St(e_0)$ such that $p(\tilde\al)=f(\al)$. If $g$ +exists, $g(\al)$ must be $\tilde\al$ and therefore $g(x)$ must be the target +$T(f(\al))(e_0)$ of $\tilde\al$. The inclusion $f(\pi(\sX,x_0))\subset p(\pi(\sE,e_0))$ +ensures that $T(f(\al))(e_0)$ is independent of the choice of $\al$, so that $g$ so +specified is a well defined functor. In fact, given another map $\al': x_0\rtarr x$, +$\al^{-1}\com\al'$ is an element of $\pi(\sX,x_0)$. Therefore +$$f(\al)^{-1}\com f(\al') = f(\al^{-1}\com\al') = p(\be)$$ +for some $\be\in \pi(\sE,e_0)$. Thus +$$p(\tilde\al\com \be) = f(\al)\com p(\be) = f(\al)\com f(\al)^{-1}\com f(\al') = f(\al').$$ +This means that $\tilde\al\com\be$ is the unique element $\tilde\al'$ of $St(e_0)$ such that +$p(\tilde\al')=f(\al')$, and its target is the target of $\tilde\al$, as required. +\end{proof} + +\begin{defn} +A map $g: \sE\rtarr \sE'$ of coverings of $\sB$ is a functor $g$ such that the +following diagram of functors is commutative: +$$\diagram +\sE \drto_p \rrto^g & & \sE' \dlto^{p'}\\ +& \sB. & \\ +\enddiagram$$ +Let Cov$(\sB)$\index{CovBb@Cov$(\sB)$} denote the category of coverings of $\sB$; when $\sB$ +is understood, we write Cov$(\sE,\sE')$ for the set of maps $\sE \rtarr \sE'$ of coverings +of $\sB$. +\end{defn} + +\begin{lem} +A map $g:\sE\rtarr \sE'$ of coverings is itself a covering. +\end{lem} +\begin{proof} +The functor $g$ is surjective on objects since, if $e'\in \sE'$ and we choose an object +$e\in \sE$ and a map $f: g(e)\rtarr e'$ in $\sE'$, then $e'= g(T(p'(f))(e))$. +The map $g: St_{\sE}(e)\rtarr St_{\sE'}(g(e))$ is a bijection since its +composite with the bijection $p': St_{\sE'}(g(e))\rtarr St_{\sB}(p'(g(e)))$ +is the bijection $p: St_{\sE}(e)\rtarr St_{\sB}(p(e))$. +\end{proof} + +The fundamental theorem immediately determines all maps of coverings of $\sB$ in +terms of group level data. + +\begin{thm} +Let $p:\sE\rtarr \sB$ and $p': \sE'\rtarr \sB$ be coverings and choose base objects +$b\in \sB$, $e\in \sE$, and $e'\in \sE'$ such that $p(e)=b=p'(e')$. There exists a map +$g:\sE\rtarr \sE'$ of coverings with $g(e)=e'$ if and only if +$$p(\pi(\sE,e))\subset p'(\pi(\sE',e')),$$ +and there is then only one such $g$. In particular, two maps of covers $g,g': \sE\rtarr \sE'$ +coincide if $g(e)=g'(e)$ for any one object $e\in \sE$. Moreover, $g$ is an isomorphism if and +only if the displayed inclusion of subgroups of $\pi(\sB,b)$ is an equality. Therefore +$\sE$ and $\sE'$ are isomorphic if and only if $p(\pi(\sE,e))$ and $p'(\pi(\sE',e'))$ are +conjugate whenever $p(e)=p'(e')$. +\end{thm} + +\begin{cor} If it exists, the universal cover of $\sB$ is unique up to isomorphism and +covers any other cover. +\end{cor} + +That the universal cover does exist will be proved in the next section. +It is useful to recast the previous theorem in terms of actions on fibers. + +\begin{thm} Let $p:\sE\rtarr \sB$ and $p': \sE'\rtarr \sB$ be coverings, choose a base +object $b\in \sB$, and let $G=\pi(\sB,b)$. If $g: \sE\rtarr \sE'$ is a map of coverings, +then $g$ restricts to a map $F_b\rtarr F'_b$ of $G$-sets, and restriction to fibers +specifies a bijection between {\em Cov}$(\sE,\sE')$ and the set of $G$-maps $F_b\rtarr F'_{b}$. +\end{thm} +\begin{proof} +Let $e\in F_b$ and $f\in \pi(\sB,b)$. By definition, $fe$ is the target of the map +$\tilde f\in St_{\sE}(e)$ such that $p(\tilde f)=f$. Clearly $g(fe)$ is the target +of $g(\tilde f)\in St_{\sE'}(g(e))$ and $p'(g(\tilde f))= p(\tilde f) = f$. Again +by definition, this gives $g(fe) = fg(e)$. The previous theorem shows that restriction +to fibers is an injection on Cov$(\sE,\sE')$. To show surjectivity, let $\al: F_b\rtarr F'_{b}$ +be a $G$-map. Choose $e\in F_b$ and let $e'=\al(e)$. Since $\al$ is a $G$-map, the isotropy +group $p(\pi(\sE,e))$ of $e$ is contained in the isotropy group $p'(\pi(\sE',e'))$ of $e'$. +Therefore the previous theorem ensures the existence of a covering map $g$ that restricts to +$\al$ on fibers. +\end{proof} + +\begin{defn} Let Aut$(\sE)\subset$\, Cov$(\sE,\sE)$\index{AutEb@Aut($\sE$)} denote the group of +automorphisms of a cover $\sE$. Note that, since it is possible to have conjugate subgroups +$H$ and $H'$ of a group $G$ such that $H$ is a proper subgroup of $H'$, it is possible to have +a map of covers $g: \sE \rtarr \sE$ such that $g$ is not an isomorphism. +\end{defn} + +\begin{cor} +Let $p:\sE\rtarr \sB$ be a covering and choose objects $b\in \sB$ and $e\in F_b$. Write +$G=\pi(\sB,b)$ and $H=p(\pi(\sE,e))$. Then {\em Aut}$(\sE)$ is isomorphic to the group of +automorphisms of the $G$-set $F_b$ and therefore to the group $WH$. If $p$ is +regular,\index{covering!regular} then {\em Aut}$(\sE)\iso G/H$. If $p$ is +universal,\index{covering!universal} then {\em Aut}$(\sE)\iso G$. +\end{cor} + +\section{The construction of coverings of groupoids} + +We have given an algebraic classification of all possible covers of $\sB$: there +is at most one isomorphism class of covers corresponding to each conjugacy class of +subgroups of $\pi(\sB,b)$. We show that all of these possibilities are actually +realized. Since this algebraic result is not needed in the proof of its topological +analogue, we shall not give complete details. + +\begin{thm} Choose a base object $b$ of $\sB$ and let $G=\pi(\sB,b)$. There is a functor +$$\sE(-): \sO(G)\rtarr \text{\em{Cov}}(\sB)$$ +that is an equivalence of categories. For each subgroup $H$ of $G$, the covering +$p: \sE(G/H)\rtarr \sB$ has a canonical base object $e$ in its fiber over $b$ such that +$$p(\pi(\sE(G/H),e)) = H.$$ +Moreover, $F_b=G/H$ as a $G$-set and, for a $G$-map $\al: G/H\rtarr G/K$ in $\sO(G)$, +the restriction of $\sE(\al): \sE(G/H)\rtarr \sE(G/K)$ to fibers over $b$ coincides +with $\al$. +\end{thm} +\begin{proof} +The idea is that, up to bijection, $St_{\sE(G/H)}(e)$ must be the same set for each $H$, +but the nature of its points can differ with $H$. At one extreme, $\sE(G/G)=\sB$, $p=\id$, +$e=b$, and the set of morphisms from $b$ to any other object $b'$ is a copy of $\pi(\sB,b)$. +At the other extreme, $\sE(G/e)$ is a universal cover of $\sB$ and there is just one +morphism from $e$ to any other object $e'$. In general, the set of objects of $\sE(G/H)$ +is defined to be $St_{\sB}(b)/H$, the coset of the identity morphism being $e$. Here $G$ +and hence its subgroup $H$ act from the right on $St_{\sB}(b)$ by composition in $\sB$. +We define $p: \sE(G/H)\rtarr \sB$ on objects by letting $p(fH)$ be the target of $f$, which +is independent of the coset representative $f$. We define morphism sets by +$$ \sE(G/H)(fH,f'H) = \sset{f'\com h\com f^{-1} | h\in H} \subset \sB(p(fH),p(f'H)).$$ +Again, this is independent of the choices of coset representatives $f$ and $f'$. Composition +and identities are inherited from those of $\sB$, and $p$ is given on morphisms by the +displayed inclusions. It is easy to check that $p: \sE(G/H)\rtarr \sB$ is a covering, +and it is clear that $p(\pi(\sE(G/H),e)) = H$. + +This defines the object function of the functor $\sE: \sO(G)\rtarr \text{Cov}(\sB)$. +To define $\sE$ on morphisms, consider $\al: G/H\rtarr G/K$. If $\al(eH) =gK$, then +$g^{-1}Hg\subset K$ and $\al(fH)=fg K$. The functor $\sE(\al):\sE(G/H)\rtarr \sE(G/K)$ +sends the object $fH$ to the object $\al(fH)=fgK$ and sends the morphism $f'\com h\com f^{-1}$ +to the same morphism of $\sB$ regarded as $f'g\com g^{-1}hg\com g^{-1}f^{-1}$. It is easily +checked that each $\sE(\al)$ is a well defined functor, and that $\sE$ is functorial in $\al$. + +To show that the functor $\sE(-)$ is an equivalence of categories, it suffices to show that +it maps the morphism set $\sO(G)(G/H,G/K)$ bijectively onto the morphism set +Cov$(\sE(G/H),\sE(G/K))$ and that every covering of $\sB$ is isomorphic to one of the +coverings $\sE(G/H)$. These statements are immediate from the results of the previous section. +\end{proof} + +The following remarks place the orbit category $\sO(\pi(\sB,b))$ in perspective by relating +it to several other equivalent categories. + +\begin{rem} Consider the category $\sS^{\sB}$ of functors $T:\sB\rtarr \sS$ and natural +transformations. Let $G=\pi(\sB,b)$. Regarding $G$ as a category with one object $b$, it is +a skeleton of $\sB$, hence the inclusion $G \subset \sB$ is an equivalence of categories. +Therefore, restriction of functors $T$ to $G$-sets $T(b)$ gives an equivalence of categories +from $\sS^{\sB}$ to the category of $G$-sets. This restricts to an equivalence between the +respective subcategories of transitive objects. We have chosen to focus on transitive objects +since we prefer to insist that coverings be connected. The inclusion of the orbit category +$\sO(G)$ in the category of transitive $G$-sets is an equivalence of categories because +$\sO(G)$ is a full subcategory that contains a skeleton. We could shrink $\sO(G)$ to a +skeleton by choosing one $H$ in each conjugacy class of subgroups of $G$, but the resulting +equivalent subcategory is a less natural mathematical object. +\end{rem} + +\section{The classification of coverings of spaces} + +In this section and the next, we shall classify covering spaces and their maps by arguments +precisely parallel to those for covering groupoids in the previous sections. In fact, applied +to the associated coverings of fundamental groupoids, some of the algebraic results directly +imply their topological analogues. We begin with the following result, which deserves +to be called the fundamental theorem of covering space theory and has many other +applications. It asserts that the fundamental group gives the only ``obstruction'' +to solving a certain lifting problem. Recall our standing assumption that all +given spaces are connected and locally path connected.\index{fundamental +theorem!of covering space theory} + +\begin{thm} +Let $p: E\rtarr B$ be a covering and let $f: X\rtarr B$ be a continuous map. +Choose $x\in X$, let $b=f(x)$, and choose $e\in F_{b}$. There exists +a map $g: X\rtarr E$ such that $g(x)=e$ and $p\com g= f$ if and only if +$$f_*(\pi_1(X,x))\subset p_*(\pi_1(E,e))$$ +in $\pi_1(B,b)$. When this condition holds, there is a unique such map $g$. +\end{thm} +\begin{proof} +If $g$ exists, its properties directly imply that $\im(f_*)\subset\im(p_*)$. +Thus assume that $\im(f_*)\subset\im(p_*)$. Applied to the covering +$\PI(p): \PI(E)\rtarr \PI(B)$, the analogue for groupoids gives a functor +$\PI(X)\rtarr \PI(E)$ that restricts on objects to the unique map $g: X\rtarr E$ +of sets such that $g(x)=e$ and $p\com g= f$. We need only check that $g$ is +continuous, and this holds because $p$ is a local homeomorphism. In detail, if +$y\in X$ and $g(y)\in U$, where $U$ is an open subset of $E$, then there is a smaller +open neighborhood $U'$ of $g(y)$ that $p$ maps homeomorphically onto an open subset $V$ +of $B$. If $W$ is any path connected neighborhood of $y$ such that $f(W)\subset V$, +then $g(W)\subset U'$ by inspection of the definition of $g$. +\end{proof} + +\begin{defn} +A map $g: E\rtarr E'$ of coverings over $B$ is a map $g$ such that the following +diagram is commutative: +$$\diagram +E \drto_p \rrto^g & & E' \dlto^{p'}\\ +& B. & \\ +\enddiagram$$ +Let Cov$(B)$\index{CovBa@Cov$(B)$} denote the category of coverings of the space $B$; when $B$ +is understood, +we write Cov$(E,E')$ for the set of maps $E \rtarr E'$ of coverings of $B$. +\end{defn} + +\begin{lem} +A map $g:E\rtarr E'$ of coverings is itself a covering. +\end{lem} +\begin{proof} +The map $g$ is surjective by the algebraic analogue. The fundamental neighborhoods +for $g$ are the components of the inverse images in $E'$ of the neighborhoods of +$B$ which are fundamental for both $p$ and $p'$. +\end{proof} + +The following remarkable theorem is an immediate consequence of the +fundamental theorem of covering space theory. + +\begin{thm} +Let $p:E\rtarr B$ and $p': E'\rtarr B$ be coverings and choose $b\in B$, $e\in E$, and +$e'\in E'$ such that $p(e)=b=p'(e')$. +There exists a map $g:E\rtarr E'$ of coverings with $g(e)=e'$ if and only if +$$p_*(\pi_1(E,e))\subset p'_*(\pi_1(E',e')),$$ +and there is then only one such $g$. In particular, two maps of covers $g,g': E\rtarr E'$ +coincide if $g(e)=g'(e)$ for any one $e\in E$. Moreover, $g$ is a homeomorphism if and only +if the displayed inclusion of subgroups of $\pi_1(B,b)$ is an equality. Therefore $E$ and +$E'$ are homeomorphic if and only if $p_*(\pi_1(E,e))$ and $p'_*(\pi_1(E',e'))$ are +conjugate whenever $p(e)=p'(e')$. +\end{thm} + +\begin{cor} If it exists, the universal cover of $B$ is unique up to isomorphism and +covers any other cover. +\end{cor} + +Under a necessary additional hypothesis on $B$, we shall prove in the next section that +the universal cover does exist. + +We hasten to add that the theorem above is atypical of algebraic topology. It is not +usually the case that algebraic invariants like the fundamental group totally determine +the existence and uniqueness of maps of topological spaces with prescribed properties. +The following immediate implication of the theorem gives one explanation. + +\begin{cor} The fundamental groupoid functor induces a bijection +$$\text{\em Cov}(E,E') \rtarr \text{\em Cov}(\PI(E),\PI(E')).$$ +\end{cor} + +Just as for groupoids, we can recast the theorem in terms of fibers. In fact, via the +previous corollary, the following result is immediate from its analogue for groupoids. + +\begin{thm} Let $p:E\rtarr B$ and $p': E'\rtarr B$ be coverings, choose a basepoint +$b\in B$, and let $G=\pi_1(B,b)$. If $g: E\rtarr E'$ is a map of coverings, +then $g$ restricts to a map $F_b\rtarr F'_b$ of $G$-sets, and restriction to fibers +specifies a bijection between {\em Cov}$(E,E')$ and the set of $G$-maps $F_b\rtarr F'_{b}$. +\end{thm} + +\begin{defn} Let Aut$(E)\subset$\,Cov$(E,E)$\index{AutEb@Aut($\sE$)} denote the group of +automorphisms of a cover $E$. Again, just as for groupoids, it is possible to have +a map of covers $g: E \rtarr E$ such that $g$ is not an isomorphism. +\end{defn} + +\begin{cor} +Let $p:E\rtarr B$ be a covering and choose $b\in B$ and $e\in F_b$. Write $G=\pi_1(B,b)$ +and $H=p_*(\pi_1(E,e))$. Then {\em Aut}$(E)$ is isomorphic to the group of automorphisms of the +$G$-set $F_b$ and therefore to the group $WH$. If $p$ is regular,\index{covering!regular} +then {\em Aut}$(E)\iso G/H$. If $p$ is universal,\index{covering!universal} then +{\em Aut}$(E)\iso G$. +\end{cor} + +\section{The construction of coverings of spaces} + +We have now given an algebraic classification of all possible covers of $B$: there +is at most one isomorphism class of covers corresponding to each conjugacy class of +subgroups of $\pi_1(B,b)$. We show here that all of these possibilities are actually +realized. We shall first construct universal covers and then show that the existence +of universal covers implies the existence of all other possible covers. Again, while +it suffices to think in terms of locally contractible spaces, appropriate generality +demands a weaker hypothesis. We say that a space $B$ is semi-locally simply +connected\index{semi-locally simply connected space} if every point $b\in B$ has a +neighborhood $U$ such that $\pi_1(U,b)\rtarr \pi_1(B,b)$ is the trivial homomorphism. + +\begin{thm} +If $B$ is connected, locally path connected, and semi-locally simply connected, +then $B$ has a universal cover.\index{covering!universal} +\end{thm} +\begin{proof} +Fix a basepoint $b\in B$. We turn the properties of paths that must hold in +a universal cover into a construction. Define $E$ to be the set of equivalence +classes of paths $f$ in $B$ that start at $b$ and define $p:E\rtarr B$ by +$p[f]=f(1)$. Of course, the equivalence relation is homotopy through paths from +$b$ to a given endpoint, so that $p$ is well defined. Thus, as a set, $E$ is +just $St_{\PI(B)}(b)$, exactly as in the construction of the universal cover of $\PI(B)$. +The topology of $B$ has a basis consisting of path connected open subsets $U$ such that +$\pi_1(U,u)\rtarr \pi_1(B,u)$ is trivial for all $u\in U$. Since every loop in $U$ is +equivalent in $B$ to the trivial loop, any two paths $u\rtarr u'$ in such a $U$ are +equivalent in $B$. We shall topologize $E$ so that $p$ is a cover with these $U$ as +fundamental neighborhoods. For a path $f$ in $B$ that starts at $b$ and ends in $U$, +define a subset $U[f]$ of $E$ by +$$U[f]=\{[g]\,|\,[g]=[c\cdot f]\ \text{for some}\ c:I\rtarr U\}.$$ +The set of all such $U[f]$ is a basis for a topology on $E$ since if $U[f]$ and +$U'[f']$ are two such sets and $[g]$ is in their intersection, then +$$W[g]\subset U[f]\cap U'[f']$$ +for any open set $W$ of $B$ such that $p[g]\in W\subset U\cap U'$. For $u\in U$, +there is a unique $[g]$ in each $U[f]$ such that $p[g]=u$. Thus $p$ maps $U[f]$ +homeomorphically onto $U$ and, if we choose a basepoint $u$ in $U$, then $p^{-1}(U)$ +is the disjoint union of those $U[f]$ such that $f$ ends at $u$. It only remains to show +that $E$ is connected, locally path connected, and simply connected, and the second of +these is clear. Give $E$ the basepoint $e=[c_b]$. For $[f]\in E$, define a path +$\tilde{f}:I\rtarr E$ by $\tilde{f}(s)=[f_s]$, where $f_s(t)=f(st)$; $\tilde{f}$ is +continuous since each $\tilde{f}^{-1}(U[g])$ is open by the definition of $U[g]$ and +the continuity of $f$. Since $\tilde{f}$ starts at $e$ and ends at $[f]$, $E$ is path +connected. Since $f_s(1)=f(s)$, $p\com\tilde{f}=f$. Thus, by definition, +$$T[f](e)=[\tilde{f}(1)]=[f].$$ +Restricting attention to loops $f$, we see that $T[f](e)=e$ if and only if $[f]=e$ +as an element of $\pi_1(B,b)$. Thus the action of $\pi_1(B,b)$ on $F_b$ is free and +the isotropy group $p_*(\pi_1(E,e))$ is trivial. +\end{proof} + +We shall construct general covers by passage to orbit spaces from the universal cover, +and we need some preliminaries. + +\begin{defn} A $G$-space $X$ is a space $X$ that is a $G$-set with continuous +action map $G\times X\rtarr X$. Define the orbit space\index{orbit space} $X/G$ to +be the set of orbits $\sset{Gx|x\in X}$ with its topology as a quotient space of $X$. +\end{defn} + +The definition makes sense for general topological groups $G$. However, our interest +here is in discrete groups $G$, for which the continuity condition just means that +action by each element of $G$ is a homeomorphism. The functoriality on $\sO(G)$ of +our construction of general covers will be immediate from the following observation. + +\begin{lem} Let $X$ be a $G$-space. Then passage to orbit spaces +defines a functor $X/(-): \sO(G)\rtarr \sU$. +\end{lem} +\begin{proof} +The functor sends $G/H$ to $X/H$ and sends a map $\al: G/H\rtarr G/K$ +to the map $X/H\rtarr X/K$ that sends the coset $Hx$ to the coset $K\ga^{-1}x$, +where $\al$ is given by the subconjugacy relation $\ga^{-1}H\ga\subset K$. +\end{proof} + +The starting point of the construction of general covers is the following +description of regular covers and in particular of the universal cover. + +\begin{prop} +Let $p: E\rtarr B$ be a cover such that {\em Aut}$(E)$ acts transitively on $F_b$. Then the +cover $p$ is regular and $E/$\,{\em Aut}$(E)$ is homeomorphic to $B$. +\end{prop} +\begin{proof} For any points $e,e'\in F_b$, there exists $g\in \text{Aut}(E)$ +such that $g(e)=e'$ and thus $p_*(\pi_1(E,e))=p_*(\pi_1(E,e'))$. Therefore all conjugates of +$p_*(\pi_1(E,e))$ are equal to $p_*(\pi_1(E,e))$ and $p_*(\pi_1(E,e))$ is a normal subgroup +of $\pi_1(B,b)$. The homeomorphism is clear since, locally, both $p$ and passage +to orbits identify the different components of the inverse images of fundamental +neighborhoods. +\end{proof} + +\begin{thm} Choose a basepoint $b\in B$ and let $G=\pi_1(B,b)$. There is a functor +$$E(-):\sO(G)\rtarr \text{\em Cov}(B)$$ +that is an equivalence of categories. For each subgroup $H$ of $G$, the covering +$p: E(G/H)\rtarr B$ has a canonical basepoint $e$ in its fiber over $b$ such that +$$p_*(\pi_1(E(G/H),e)) = H.$$ +Moreover, $F_b\iso G/H$ as a $G$-set and, for a $G$-map $\al: G/H\rtarr G/K$ in $\sO(G)$, +the restriction of $E(\al): E(G/H)\rtarr E(G/K)$ to fibers over $b$ coincides with $\al$. +\end{thm} +\begin{proof} Let $p: E\rtarr B$ be the universal cover of $B$ and fix $e\in E$ such that $p(e)=b$. +We have the isomorphism Aut$(E)\iso\pi_1(B,b)$ given by mapping $g: E\rtarr E$ to the path +class $[f]\in G$ such that $g(e)=T(f)(e)$, where $T(f)(e)$ is the endpoint of the path +$\tilde f$ that starts at $e$ and lifts $f$. We identify subgroups of $G$ with subgroups of +Aut$(E)$ via this isomorphism. We define $E(G/H)$ to be the orbit space $E/H$ and we let +$q: E\rtarr E/H $ be the quotient map. We may identify $B$ with $E/\text{Aut}(E)$, and +inclusion of orbits specifies a map $p': E/H\rtarr B$ such that $p'\com q=p: E\rtarr B$. +If $U\subset B$ is a fundamental neighborhood for $p$ and $V$ is a component of +$p^{-1}(U)\subset E$, then +$$p^{-1}(U)=\textstyle{\coprod}_{g\in \text{Aut}(E)}\, gV.$$ +Passage to orbits over $H$ simply identifies some of these components, and we see +immediately that both $p'$ and $q$ are covers. If $e'=q(e)$, then $p_*'$ maps +$\pi_1(E/H,e')$ isomorphically onto $H$ since, by construction, the isotropy group +of $e'$ under the action of $\pi_1(B,b)$ is precisely $H$. Rewriting $p'=p$ and $e'=e$ +generically, this gives the stated properties of the coverings $E(G/H)$. +The functoriality on $\sO(G)$ follows directly from the previous lemma. + +The functor $E(-)$ is an equivalence of categories since the results of the previous +section imply that it maps the morphism set $\sO(G)(G/H,G/K)$ bijectively onto the +morphism set Cov$(E(G/H),E(G/K))$ and that every covering of $B$ is isomorphic +to one of the coverings $E(G/H)$. +\end{proof} + +The classification theorems for coverings of spaces and coverings of groupoids are +nicely related. In fact, the following diagram of functors commutes up to natural +isomorphism: +$$\diagram + & \sO(\pi_1(B,b))\dlto_{E(-)} \drto^{\sE(-)} & \\ +\text{Cov}(B) \rrto_{\PI} & & \text{Cov}(\PI(B)). \\ +\enddiagram$$ + +\begin{cor} $\PI: \text{\em Cov}(B) \rtarr \text{\em Cov}(\PI(B))$ is an equivalence of categories. +\end{cor} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} + +In the following two problems, let $G$ be a connected and locally path connected topological +group\index{topological group} with identity element $e$, let $p: H\rtarr G$ be a covering, +and fix $f\in H$ such +that $p(f)=e$. Prove the following. (Hint: Make repeated use of the fundamental theorem +for covering spaces.) +\begin{enumerate} +\item +\begin{enumerate} +\item[(a)] $H$ has a unique continuous product $H\times H\rtarr H$ with identity element $f$ +such that $p$ is a homomorphism. +\item[(b)] $H$ is a topological group under this product, and $H$ is Abelian if $G$ is. +\end{enumerate} +\item +\begin{enumerate} +\item[(a)] The kernel $K$ of $p$ is a discrete normal subgroup of $H$. +\item[(b)] In general, any discrete normal subgroup $K$ of a connected topological group $H$ +is contained in the center of $H$. +\item[(c)] For $k\in K$, define $t(k): H\rtarr H$ by $t(k)(h) = kh$. Then $k\rtarr t(k)$ +specifies an isomorphism between $K$ and the group Aut$(H)$. +\end{enumerate} +\end{enumerate} + +Let $X$ and $Y$ be connected, locally path connected, and Hausdorff. A map $f: X\rtarr Y$ +is said to be a local homeomorphism\index{local homeomorphism} +if every point of $X$ has an open neighborhood that maps +homeomorphically onto an open set in $Y$. + +\begin{enumerate} +\item[3.] Give an example of a surjective local homeomorphism that is not a covering. +\item[4.]* Let $f: X\rtarr Y$ be a local homeomorphism, where $X$ is compact. Prove that $f$ +is a (surjective!) covering with finite fibers. +\end{enumerate} + +Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, +define +$$X^H=\sset{x|hx = x\ \text{for all}\ \ h\in H}\subset X;$$ +$X^H$ is the $H$-fixed point subspace\index{fixed point space} of $X$. +Topologize the set of functions $G/H\rtarr X$ as the product of copies +of $X$ indexed on the elements of $G/H$, and give the set of $G$-maps +$G/H\rtarr X$ the subspace topology. + +\begin{enumerate} +\item[5.] Show that the space of $G$-maps $G/H\rtarr X$ +is naturally homeomorphic to $X^H$. In particular, $\sO(G/H,G/K)\iso (G/K)^H$. +\item[6.] Let $X$ be a $G$-space. Show that passage to fixed point spaces, +$G/H \longmapsto X^H$, is the object function of a {\em contravariant} +functor $X^{(-)}: \sO(G)\rtarr \sU$. +\end{enumerate} + +\chapter{Graphs} + +We define graphs, describe their homotopy types, and use them to +show that a subgroup of a free group is free and that any group +is the fundamental group of some space. + +\section{The definition of graphs} + +We give the definition in a form that will later make it clear that a +graph is exactly a one-dimensional CW complex. Note that the zero-sphere +$S^0$ is a discrete space with two points. We think of $S^0$ as +the boundary of $I$ and so label the points $0$ and $1$. + +\begin{defn} A graph\index{graph} $X$ is a space that is obtained from a (discrete) set $X^0$ +of points, called vertices\index{vertex}, and a (discrete) set $J$ of functions +$j: S^0\rtarr X^0$ +as the quotient space of the disjoint union $X^0\amalg (J\times I)$ that is +obtained by identifying $(j,0)$ with $j(0)$ and $(j,1)$ with $j(1)$. +The images of the intervals $\{ j\}\times I$ are called edges\index{edge}. A graph is +finite if it has only finitely many vertices and edges or, equivalently, if +it is a compact space. A graph is locally finite if each vertex is a boundary +point of only finitely many edges or, equivalently, if it is a locally compact +space. A subgraph $A$ of $X$ is a graph $A\subset X$ with $A^0\subset X^0$. +That is, $A$ is the union of some of the vertices and edges of $X$. +\end{defn} + +Observe that a graph is a locally contractible\index{locally contractible space} +space: any neighborhood of any point contains a contractible neighborhood of +that point. Therefore a connected graph has all possible covers. + +\section{Edge paths and trees} + +An oriented edge $k:I\rtarr X$ in a graph $X$ is the traversal of an edge in +either the forward or backward direction. An edge path\index{edge path} is a finite +composite of +oriented edges $k_n$ with $k_{n+1}(0)=k_n(1)$. Such a path is reduced\index{edge path!reduced} +if it is never the case that $k_{n+1}$ is $k_n$ with the opposite orientation. An edge +path is closed\index{edge path!closed} if it starts and ends at the same vertex +(and is thus a loop). + +\begin{defn} A tree\index{tree} is a connected graph with no closed reduced edge paths. +\end{defn} + +A subspace $A$ of a space $X$ is a deformation retract\index{deformation retract} if +there is a homotopy +$h:X\times I\rtarr X$ such that $h(x,0)=x$, $h(a,t)=a$, and $h(x,1)\in A$ +for all $x\in X$, $a\in A$, and $t\in I$. Such a homotopy is called a +deformation\index{deformation} of $X$ onto $A$. + +\begin{lem} +Any vertex $v_0$ of a tree $T$ is a deformation retract of $T$. +\end{lem} +\begin{proof} +This is true by induction on the number of edges when $T$ is finite since +we can prune the last branch. For the general case, observe that each vertex +$v$ lies in some finite connected subtree $T(v)$ that also contains $v_0$. +Choose an edge path $a(v):I\rtarr T(v)$ connecting $v$ to $v_0$. For an edge +$j$ from $v$ to $v'$, $T(v)\cup T(v')\cup j$ is a finite connected subtree of $T$. +On the square $j\times I$, we define +$$h: j\times I\rtarr T(v)\cup T(v')\cup j$$ +by requiring $h=a(v)$ on $\sset{v}\times I$, $h=a(v')$ on +$\sset{v'}\times I$, $h(x,0)=x$ and $h(x,1)=v_0$ for all +$x\in j$, and extending over the interior of the square by use of the +simple connectivity of $T(v)\cup T(v')\cup j$. As $j$ runs over the edges, +these homotopies glue together to specify a deformation $h$ of $T$ onto $v_0$. +\end{proof} + +A subtree of a graph $X$ is maximal{\index{tree!maximal} if it is contained in no +strictly larger tree. + +\begin{lem} If a tree $T$ is a subgraph of a graph $X$, then $T$ is contained in +a maximal tree. If $X$ is connected, then a tree in $X$ is maximal if and only if +it contains all vertices of $X$. +\end{lem} +\begin{proof} +Since the union of an increasing family of trees in $X$ is a tree, the first +statement holds by Zorn's lemma. If $X$ is connected, then a tree containing all +vertices is maximal since addition of an edge would result in a subgraph that +contains a closed reduced edge path and, conversely, a tree $T$ that does not +contain all vertices is not maximal since a vertex not in $T$ can be connected +to a vertex in $T$ by a reduced edge path consisting of edges not in $T$. +\end{proof} + +\section{The homotopy types of graphs} + +Graph theory is a branch of combinatorics. The homotopy theory of graphs is +essentially trivial, by the following result. + +\begin{thm} +Let $X$ be a connected graph with maximal tree $T$. Then the quotient space +$X/T$ is the wedge of one circle for each edge of $X$ not in $T$, and the +quotient map $q: X\rtarr X/T$ is a homotopy equivalence. +\end{thm} +\begin{proof} +The first clause is evident. The second is a direct consequence of a later +result (that will be left as an exercise): for a suitably nice inclusion, +called a ``cofibration,'' of a contractible space $T$ in a space $X$, the +quotient map $X\rtarr X/T$ is a homotopy equivalence. A direct proof in the +present situation is longer and uglier. With the notation in our proof that +a vertex $v_0$ is a deformation retract of $T$ via a deformation $h$, +define a loop $b_j=a(v')\cdot j\cdot a(v)^{-1}$ at $v_0$ for each edge $j: v\rtarr v'$ not +in $T$. The $b_j$ together specify a map $b$ from $X/T\iso\bigvee_j S^1$ to $X$. +The composite $q\com b: X/T\rtarr X/T$ is the wedge over $j$ of copies of the loop +$c_{v_0}\cdot \id \cdot\, c_{v_0}^{-1}:S^1\rtarr S^1$ and is therefore homotopic to the identity. +To prove that $b\com q$ is homotopic to the identity, observe that $h$ +is a homotopy $\id\htp b\com q$ on $T$. This homotopy extends to a homotopy +$H:\id\htp b\com q$ on all of $X$. To see this, we need only construct $H$ on +$j\times I$ for an edge $j:v\rtarr v'$ not in $T$. The following schematic +description of the prescribed behavior on the boundary of the square makes it +clear that $H$ exists: +$$\diagram +\xline[0,3]^<(0.2){a(v)^{-1}} ^<(0.5){j} ^<(0.8){a(v')} \xline[3,0]_{a(v)} +& \xline[3,-1]^{c_v} & \xline[3,1]_{c_{v'}} & \xline[3,0]^{a(v')} \\ +& & & \\ +& & & \\ +\xline[0,3]_{j} & & & \\ +\enddiagram$$ +\renewcommand{\qed}{}\end{proof} + +\section{Covers of graphs and Euler characteristics} + +Define the Euler characteristic\index{Euler characteristic! of a finite graph} $\ch (X)$ of a +finite graph $X$ to be $V-E$, where +$V$ is the number of vertices of $X$ and $E$ is the number of edges. By induction +on the number of edges, $\ch (T)=1$ for any finite tree. The determination of the +homotopy types of graphs has the following immediate implication. + +\begin{cor} If $X$ is a connected graph, then $\pi_1(X)$ is a free group with one +generator for each edge not in a given maximal tree. If $X$ is finite, then $\pi_1(X)$ +is free on $1-\ch (X)$ generators; in particular, $\ch (X)\leq 1$, with equality if and +only if $X$ is a tree. +\end{cor} + +\begin{thm} +If $B$ is a connected graph with vertex set $B^0$ and $p:E\rtarr B$ is a covering, +then $E$ is a connected graph with vertex set $E^0=p^{-1}(B^0)$ and with one edge for +each edge $j$ of $B$ and point $e\in F_{j(0)}$. Therefore, if $B$ is finite +and $p$ is a finite cover whose fibers have cardinality $n$, then $E$ is finite and +$\ch (E)= n\ch (B)$. +\end{thm} +\begin{proof} Regard an edge $j$ of $B$ as a path $I\rtarr B$ and let $k(e):I\rtarr E$ +be the unique path such that $p\com k = j$ and $k(e)(0)=e$, where $e\in F_{j(0)}$. We +claim that $E$ is a graph with $E^0$ as vertex set and the $k(e)$ as edges. An easy +path lifting argument shows that each point of $E-E^0$ is an interior point of exactly +one edge, hence we have a continuous bijection from the graph $E^0\amalg (K\times I)/(\sim )$ +to $E$, where $K$ is the evident set of ``attaching maps'' $S^0\rtarr E^0$ for the +specified edges. This map is a homeomorphism since it is a local homeomorphism over $B$. +\end{proof} + +\section{Applications to groups} + +The following purely algebraic result is most simply proved by topology. + +\begin{thm} +A subgroup $H$ of a free group\index{free group} $G$ is free. If $G$ is free on +$k$ generators and $H$ has +finite index $n$ in $G$, then $H$ is free on $1-n+nk$ generators. +\end{thm} +\begin{proof} +Realize $G$ as $\pi_1(B)$, where $B$ is the wedge of one circle for each generator of $G$ +in a given free basis. Construct a covering $p: E\rtarr B$ such that $p_*(\pi_1(E))=H$. +Since $E$ is a graph, $H$ must be free. If $G$ has $k$ generators, then $\ch (B)=1-k$. If +$[G:H]=n$, then $F_b$ has cardinality $n$ and $\ch (E)= n\ch(B)$. Therefore +$1-\ch(E)=1-n+nk$. +\end{proof} + +We can extend the idea to realize any group as the fundamental group of some connected +space.\index{fundamental group} + +\begin{thm} +For any group $G$, there is a connected space $X$ such that $\pi_1(X)$ is +isomorphic to $G$. +\end{thm} +\begin{proof} +We may write $G=F/N$ for some free group $F$ and normal subgroup $N$. As above, we +may realize the inclusion of $N$ in $F$ by passage to fundamental groups from a cover +$p:E\rtarr B$. Define the (unreduced) cone on $E$ to be $CE = (E\times I)/(E\times\sset{1})$ +and define +$$X=B\cup_pCE/(\sim),$$ +where $(e,0)\sim p(e)$. Let $U$ and $V$ be the images in $X$ of $B\amalg (E\times [0,3/4))$ +and $E\times (1/4,1]$, respectively, and choose a basepoint in $E\times \sset{1/2}$. Since +$U$ and $U\cap V$ are homotopy equivalent to $B$ and $E$ via evident deformations and $V$ +is contractible, a consequence of the van Kampen theorem gives the conclusion. +\end{proof} + +The space $X$ constructed in the proof is called the ``homotopy cofiber''\index{homotopy cofiber} +of the map $p$. It is an important general construction to which we shall return shortly. + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Let $F$ be a free group on two generators $a$ and $b$. How many subgroups of $F$ have +index $2$? Specify generators for each of these subgroups. +\item Prove that a non-trivial normal subgroup $N$ with infinite index in a free group $F$ +cannot be finitely generated. +\item* Essay: Describe a necessary and sufficient condition for a graph to be embeddable +in the plane. +\end{enumerate} + +\chapter{Compactly generated spaces} + +We briefly describe the category of spaces in which algebraic topologists +customarily work. The ordinary category of spaces allows pathology that +obstructs a clean development of the foundations. The homotopy and homology +groups of spaces are supported on compact subspaces, and it turns out that if +one assumes a separation property that is a little weaker than the Hausdorff +property, then one can refine the point-set topology of spaces to eliminate +such pathology without changing these invariants. We shall leave the proofs +to the reader, but the wise reader will simply take our word for it, at least +on a first reading: we do not want to overemphasize this material, the +importance of which can only become apparent in retrospect. + +\section{The definition of compactly generated spaces} + +We shall understand compact spaces\index{compact space} to be both compact and Hausdorff, following +Bourbaki. A space $X$ is said to be ``weak Hausdorff''\index{weak Hausdorff space} if $g(K)$ is closed +in $X$ for every map $g: K\rtarr X$ from a compact space $K$ into $X$. When this holds, +the image $g(K)$ is Hausdorff and is therefore a compact subspace of $X$. This separation +property lies between $T_1$ (points are closed) and Hausdorff, but it is not much weaker +than the latter. + +A subspace $A$ of $X$ is said to be ``compactly closed''\index{compactly closed subspace} if +$g^{-1}(A)$ is closed in $K$ +for any map $g:K\rtarr X$ from a compact space $K$ into $X$. When $X$ is weak Hausdorff, +this holds if and only if the intersection of $A$ with each compact subset of $X$ is +closed. A space $X$ is a ``$k$-space''\index{kspace@$k$-space} if every compactly closed +subspace is closed. + +A space $X$ is ``compactly generated''\index{compactly generated space} if it is a weak Hausdorff +$k$-space. For example, +any locally compact space and any weak Hausdorff space that satisfies the first axiom of +countability\index{first axiom of countability} +(every point has a countable neighborhood basis) is compactly generated. We +have expressed the definition in a form that should make the following statement clear. + +\begin{lem} If $X$ is a compactly generated space and $Y$ is any space, then a function +$f:X\rtarr Y$ is continuous if and only if its restriction to each compact +subspace $K$ of $X$ is continuous. +\end{lem} + +We can make a space $X$ into a $k$-space by giving it a new topology in which a +space is closed if and only if it is compactly closed in the original topology. +We call the resulting space $kX$. Clearly the identity function $kX\rtarr X$ is +continuous. If $X$ is weak Hausdorff, then so is $kX$, hence $kX$ is compactly +generated. Moreover, $X$ and $kX$ then have exactly the same compact subsets. + +Write $X\times_c Y$ for the product of $X$ and $Y$ with its usual topology and +write $X\times Y=k(X\times_cY)$. If $X$ and $Y$ are weak Hausdorff, then +$X\times Y=kX\times kY$. If $X$ is locally compact and $Y$ is compactly generated, +then $X\times Y=X\times_cY$. + +By definition, a space $X$ is Hausdorff if the diagonal subspace $\DE X=\sset{(x,x)}$ +is closed in $X\times_c X$. The weak Hausdorff property admits a similar characterization. + +\begin{lem} If $X$ is a $k$-space, then $X$ is weak Hausdorff if and only if +$\DE X$ is closed in $X\times X$. +\end{lem} + + +\section{The category of compactly generated spaces} + +One major source of point-set level pathology can be passage to quotient spaces\index{quotient +space}. Use +of compactly generated topologies alleviates this. + +\begin{prop} +If $X$ is compactly generated and $\pi: X\rtarr Y$ is a quotient map, then $Y$ is +compactly generated if and only if $(\pi\times\pi)^{-1}(\DE Y)$ is closed in $X\times X$. +\end{prop} + +The interpretation is that a quotient space of a compactly generated space by a +``closed equivalence relation'' is compactly generated. We are particularly interested +in the following consequence. + +\begin{prop} +If $X$ and $Y$ are compactly generated spaces, $A$ is a closed subspace of $X$, and +$f: A\rtarr Y$ is any continuous map, then the pushout $Y\cup_f X$ is compactly generated. +\end{prop} + + +Another source of pathology is passage to colimits over sequences of maps $X_i\rtarr X_{i+1}$. +When the given maps are inclusions, the colimit is the union of the sets $X_i$ with the +``topology of the union;''\index{topology of the union} a set is closed if and +only if its intersection with each $X_i$ is closed. + +\begin{prop} +If $\sset{X_i}$ is a sequence of compactly generated spaces and inclusions +$X_i\rtarr X_{i+1}$ with closed images, then\, {\em colim}$\,X_i$ is compactly +generated. +\end{prop} + +We now adopt a more categorical point of view. We redefine $\sU$\index{U@$\sU$} to be the category +of compactly generated spaces and continuous maps, and we redefine $\sT$\index{T@$\sT$} to be its +subcategory of based spaces and based maps. + +Let $w\sU$ be the category of weak Hausdorff spaces. We have the functor $k: w\sU\rtarr \sU$, +and we have the forgetful functor $j:\sU\rtarr w\sU$, which embeds $\sU$ as a full subcategory +of $w\sU$. Clearly +$$\sU(X,kY)\iso w\sU(jX,Y)$$ +for $X\in\sU$ and $Y\in w\sU$ since the identity map $kY\rtarr Y$ is continuous and +continuity of maps defined on compactly generated spaces is compactly determined. +Thus $k$ is right adjoint to $j$. + +We can construct colimits and limits of spaces by performing these constructions +on sets: they inherit topologies that give them the universal properties of +colimits\index{colimit} and +limits\index{limit} in the classical category of spaces. Limits of weak Hausdorff spaces are weak +Hausdorff, but limits of $k$-spaces need not be $k$-spaces. We construct limits of compactly +generated spaces by applying the functor $k$ to their limits as spaces. It is +a categorical fact that functors which are right adjoints preserve limits, so this does give +categorical limits in $\sU$. This is how we defined $X\times Y$, for example. + +Point-set level colimits of weak Hausdorff spaces need not be weak Hausdorff. +However, if a point-set level colimit of compactly generated spaces is weak Hausdorff, +then it is a $k$-space and therefore compactly generated. We shall only be interested in +colimits in those cases where this holds. The propositions above give examples. +In such cases, these constructions give categorical colimits in $\sU$. + +From here on, we agree that all given spaces are to be compactly generated, and we +agree to redefine any construction on spaces by applying the functor $k$ to it. For +example, for spaces $X$ and $Y$ in $\sU$, we understand the function space\index{function space} +$\text{Map}(X,Y)=Y^X$ to mean the set of continuous maps from $X$ to $Y$ with the +$k$-ification of the standard compact-open topology;\index{compact-open topology} the latter +topology has as basis the +finite intersections of the subsets of the form $\{ f|f(K)\subset U\}$ for some compact subset +$K$ of $X$ and open subset $U$ of $Y$. This leads to the following adjointness homeomorphism, +which holds without restriction when we work in the category of compactly generated spaces. + +\begin{prop} +For spaces $X$, $Y$, and $Z$ in $\sU$, the canonical bijection +$$Z^{(X\times Y)} \iso (Z^Y)^X$$ +is a homeomorphism. +\end{prop} + +Observe in particular that a homotopy $X\times I\rtarr Y$ can equally well be +viewed as a map $X\rtarr Y^I$. These adjoint, or ``dual,'' points of view will +play an important role in the next two chapters. + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item +\begin{enumerate} +\item[(a)] Any subspace of a weak Hausdorff space is weak Hausdorff. +\item[(b)] Any closed subspace of a $k$-space is a $k$-space. +\item[(c)] An open subset $U$ of a compactly generated space $X$ is compactly +generated if each point has an open neighborhood in $X$ with closure contained +in $U$. +\end{enumerate} +\item* A Tychonoff (or completely regular) space $X$ is a $T_1$-space (points are +closed) such that for each point $x\in X$ and each closed subset $A$ such that +$x\notin A$, there is a function $f: X\rtarr I$ such that $f(x)=0$ and $f(a)=1$ if +$a\in A$. Prove the following (e.g., Kelley, {\em General Topology}). +\begin{enumerate} +\item[(a)] A space is Tychonoff\index{Tychonoff space} if and only if it can be +embedded in a cube (a product of copies of $I$). +\item[(b)] There are Tychonoff spaces that are not $k$-spaces, but every cube is +a compact Hausdorff space. +\end{enumerate} +\item Brief essay: In view of Problems 1 and 2, what should we mean by a ``subspace'' of a +compactly generated space. (We do {\em not} want to restrict the allowable set of subsets.) +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{Cofibrations} + +Exact sequences that feature in the study of homotopy, homology, and cohomology +groups all can be derived homotopically from the theory of cofiber and fiber +sequences that we present in this and the following two chapters. Abstractions of +these ideas are at the heart of modern axiomatic treatments of homotopical algebra +and of the foundations of algebraic $K$-theory. + +The theories of cofiber and fiber sequences illustrate an important, but +informal, duality theory, known as Eckmann-Hilton duality.\index{Eckmann-Hilton duality} It +is based on the +adjunction between Cartesian products and function spaces. Our standing +hypothesis that all spaces in sight are compactly generated allows the +theory to be developed without further restrictions on the given spaces. +We discuss ``cofibrations'' here and the ``dual'' notion of ``fibrations'' +in the next chapter. + +\section{The definition of cofibrations} + +\begin{defn} +A map $i: A\rtarr X$ is a cofibration\index{cofibration} if it satisfies the homotopy extension +property (HEP).\index{homotopy extension property}\index{HEP} This means that if $h\com i_0=f\com i$ in the +diagram +$$\diagram +A \rrto^{i_0} \ddto_{i} & & A\times I \dlto_{h} \ddto^{i\times\id} \\ +& Y & \\ +X \rrto_{i_0} \urto^{f} & & X\times I, \uldashed_{\tilde{h}}|>\tip\\ +\enddiagram$$ +then there exists $\tilde{h}$ that makes the diagram commute. +\end{defn} + +Here $i_0(x)=(x,0)$. We do not require $\tilde{h}$ to be unique, and it +usually isn't. Using our alternative way of writing homotopies, we see that +the ``test diagram'' displayed in the definition can be rewritten in the +equivalent form +$$\diagram +A \dto_i \rto^{h} & Y^I \dto^{p_0} \\ +X \urdashed^{\tilde{h}}|>\tip \rto_f & Y, \\ +\enddiagram$$ +where $p_0(\xi)=\xi(0)$. + +Pushouts\index{pushout} of cofibrations are cofibrations, in the sense +of the following result. We generally write $B\cup_g X$ for the pushout +of a given cofibration $i:A\rtarr X$ and a map $g:A\rtarr B$. + +\begin{lem} +If $i:A\rtarr X$ is a cofibration and $g :A\rtarr B$ is any map, then +the induced map $B\rtarr B\cup_g X$ is a cofibration. +\end{lem} +\begin{proof} +Notice that $(B\cup_g X)\times I\iso (B\times I)\cup_{g\times \id}(X\times I)$ +and consider a typical test diagram for the HEP. The proof is a formal +chase of the following diagram: +$$\diagram +A \xto[0,4]^{i_0} \xto[4,0]^{\ \ \text{pushout}}_{i} \drto^g & & & & +A\times I \xto[4,0]^{i\times\id}_{\text{pushout}\ \ \ \, } \dlto_{g\times \id}\\ +& B \ddto \rrto & & B\times I \dlto_h \ddto & \\ +& & Y & & \\ +& B\cup_g X \urto^{f} \rrto +& &(B\cup_g X)\times I \uldashed_{\tilde{h}}|>\tip & \\ +X \urto \xto[0,4]_{i_0} & & & & X\times I. \ulto +\xdashed '[-1,-2]^{\bar{h}} '[-2,-2]|>\tip \\ +\enddiagram$$ +We first use that $A\rtarr X$ is a cofibration to obtain a homotopy +$\bar{h}: X\times I\rtarr Y$ and then use the right-hand pushout to +see that $\bar{h}$ and $h$ induce the required homotopy $\tilde{h}$. +\end{proof} + +\section{Mapping cylinders and cofibrations} + +Although the HEP is expressed in terms of general test diagrams, there is a +certain universal test diagram. Namely, we can let $Y$ in our original test +diagram be the ``mapping cylinder''\index{mapping cylinder} +$$Mi \equiv X\cup_i(A\times I),$$ +which is the pushout of $i$ and $i_0$. Indeed, suppose that we can construct +a map $r$ that makes the following diagram commute: +$$\diagram +A \rrto^{i_0} \ddto_{i} & & A\times I \dlto \ddto^{i\times\id} \\ +& Mi & \\ +X \rrto_{i_0} \urto & & X\times I. \uldashed_{r}|>\tip\\ +\enddiagram$$ +By the universal property of pushouts, the given maps $f$ and $h$ in our original +test diagram induce a map $Mi\rtarr Y$, and its composite with $r$ gives a homotopy +$\tilde{h}$ that makes the test diagram commute. + +A map $r$ that makes the previous diagram commute satisfies $r\com j=\id$, where +$j: Mi\rtarr X\times I$ is the map that restricts to $i_0$ on $X$ and to $i\times \id$ +on $A\times I$. As a matter of point-set topology, left as an exercise, it follows that +a cofibration is an inclusion with closed image. + +\section{Replacing maps by cofibrations} + +We can use the mapping cylinder construction to decompose an arbitrary map +$f: X\rtarr Y$ as the composite of a cofibration and a homotopy equivalence. +That is, up to homotopy, any map can be replaced by a cofibration. To see this, +recall that $Mf = Y\cup_f (X\times I)$ and observe that $f$ +coincides with the composite +$$X \overto{j} Mf \overto{r} Y,$$ +where $j(x)=(x,1)$ and where $r(y)=y$ on $Y$ and $r(x,s)=f(x)$ on $X\times I$. +If $i: Y\rtarr Mf$ is the inclusion, then $r\com i=\id$ and $\id\htp i\com r$. +In fact, we can define a deformation $h: Mf\times I\rtarr Mf$ of $Mf$ onto $i(Y)$ +by setting +$$h(y,t) = y \ \ \tand \ \ h((x,s),t)=(x,(1-t)s).$$ +It is not hard to check directly that $j: X\rtarr Mf$ satisfies the HEP, and this +will also follow from the general criterion for a map to be a cofibration to which +we turn next. + +\section{A criterion for a map to be a cofibration} + +We want a criterion that allows us to recognize cofibrations when we see +them. We shall often consider pairs $(X,A)$ consisting of a space $X$ and +a subspace $A$. Co\-fibration pairs will be those pairs that ``behave +homologically'' just like the associated quotient spaces $X/A$. + +\begin{defn} +A pair $(X,A)$ is an NDR-pair\index{NDR-pair} (= neighborhood deformation retract pair) if +there is a map $u: X\rtarr I$ such that $u^{-1}(0)=A$ and a homotopy +$h:X\times I\rtarr X$ such that $h_0=\id$, $h(a,t)=a$ for $a\in A$ and +$t\in I$, and $h(x,1)\in A$ if $u(x)<1$; $(X,A)$ is a DR-pair\index{DR-pair} if $u(x)<1$ +for all $x\in X$, in which case $A$ is a deformation retract of $X$. +\end{defn} + +\begin{lem} +If $(h,u)$ and $(j,v)$ represent $(X,A)$ and $(Y,B)$ as NDR-pairs, then +$(k,w)$ represents the ``product pair'' $(X\times Y,X\times B\cup A\times Y)$ +as an NDR-pair, where $w(x,y)=\text{\em min}(u(x),v(y))$ and +$$k(x,y,t)= +\begin{cases} +(h(x,t),j(y,tu(x)/v(y))) \ \ \text{if} \ v(y)\geq u(x)\\ +(h(x,tv(y)/u(x)),j(y,t)) \ \ \text{if} \ u(x)\geq v(y). +\end{cases}$$ +If $(X,A)$ or $(Y,B)$ is a DR-pair, then so is $(X\times Y,X\times B\cup A\times Y)$. +\end{lem} +\begin{proof} +If $v(y)=0$ and $v(y)\geq u(x)$, then $u(x)=0$ and both $y\in B$ and $x\in A$; therefore +we can and must understand $k(x,y,t)$ to be $(x,y)$. It is easy to check from this and +the symmetric observation that $k$ is a well defined continuous homotopy as desired. +\end{proof} + +\begin{thm} +Let $A$ be a closed subspace of $X$. Then the following are equivalent: +\begin{enumerate} +\item[(i)] $(X,A)$ is an NDR-pair. +\item[(ii)] $(X\times I, X\times\sset{0}\cup A\times I)$ is a DR-pair. +\item[(iii)] $X\times\sset{0}\cup A\times I$ is a retract of $X\times I$. +\item[(iv)] The inclusion $i:A\rtarr X$ is a cofibration. +\end{enumerate} +\end{thm} +\begin{proof} +The lemma gives that (i) implies (ii), (ii) trivially implies (iii), and we +have already seen that (iii) and (iv) are equivalent. Assume given a retraction +$r: X\times I\rtarr X\times\sset{0}\cup A\times I$. Let +$\pi_1: X\times I\rtarr X$ and $\pi_2: X\times I\rtarr I$ be the projections +and define $u: X\rtarr I$ by +$$u(x) = \text{sup}\{ t-\pi_2r(x,t)|t\in I\}$$ +and $h: X\times I\rtarr X$ by +$$h(x,t)=\pi_1r(x,t).$$ +Then $(h,u)$ represents $(X,A)$ as an NDR-pair. Here $u^{-1}(0)=A$ since +$u(x)=0$ implies that $r(x,t)\in A\times I$ for $t>0$ and thus also for +$t=0$ since $A\times I$ is closed in $X\times I$. +\end{proof} + +\section{Cofiber homotopy equivalence} + +It is often important to work in the category of spaces under a given space $A$, and we +shall later need a basic result about homotopy equivalences in this category. We shall +also need a generalization concerning homotopy equivalences of pairs. The reader is warned +that the results of this section, although easy enough to understand, have fairly lengthy +and unilluminating proofs. + +A space +under $A$ is a map $i: A\rtarr X$. A map of spaces under $A$ is a commutative diagram +$$\diagram +& A \dlto_i \drto^{j}\\ +X\rrto_f & & Y \\ +\enddiagram$$ +A homotopy between maps under $A$ is a homotopy that at each time $t$ is a map under $A$. +We then write $h: f\htp f'\ \text{rel}\ A$ and have $h(i(a),t)=j(a)$ for all $a\in A$ and +$t\in I$. There results a notion of a homotopy equivalence under $A$. Such an equivalence +is called a ``cofiber homotopy equivalence.''\index{cofiber homotopy equivalence} The +name is suggested by the following result, +whose proof illustrates a more substantial use of the HEP than we have seen before. + +\begin{prop} Let $i: A\rtarr X$ and $j: A\rtarr Y$ be cofibrations and let $f: X\rtarr Y$ +be a map such that $f\com i = j$. Suppose that $f$ is a homotopy equivalence. Then $f$ is a +cofiber homotopy equivalence. +\end{prop} +\begin{proof} It suffices to find a map $g: Y\rtarr X$ under $A$ and a homotopy +$g\com f\htp \id \ \text{rel}\ A$. Indeed, $g$ will then be a homotopy equivalence, and +we can repeat the argument to obtain $f': X\rtarr Y$ such that $f'\com g\htp \id \ \text{rel}\ A$; +it will follow formally that $f'\htp f\ \text{rel}\ A$. +By hypothesis, there is a map $g'': Y\rtarr X$ that is a homotopy inverse to $f$. Since +$g''\com f\htp\id$, $g''\com j\htp i$. Since $j$ satisfies the HEP, it follows directly +that $g''$ is homotopic to a map $g'$ such that $g'\com j=i$. It suffices to prove that +$g'\com f: X\rtarr X$ has a left homotopy inverse $e: X\rtarr X$ under $A$, since +$g=e\com g'$ will then satisfy $g\com f\htp \id \ \text{rel}\ A$. Replacing our original +map $f$ with $g'\com f$, we see that it suffices to obtain a left homotopy inverse under $A$ +to a map $f:X\rtarr X$ such that $f\com i=i$ and $f\htp\id$. Choose a homotopy $h:f\htp \id$. +Since $h_0\com i=f\com i=i$ and $h_1=\id$, we can apply the HEP to +$h\com(i\times\id): A\times I\rtarr X$ and the identity map of $X$ to obtain a homotopy +$k:\id \htp k_1\equiv e$ such that $k\com(i\times\id)=h\com(i\times\id)$. Certainly $e\com i=i$. +Now apply the HEP to the following diagram: +$$\diagram +A\times I \rrto^{i_0} \ddto_{i\times\id} & +& A\times I\times I \ddto^{i\times\id\times\id} \dlto_K\\ +& X & \\ +X\times I \urto^J \rrto_{i_0} & & X\times I\times I. \uldashed_{L}|>\tip \\ +\enddiagram$$ +Here $J$ is the homotopy $e\com f\htp \id$ specified by +$$J(x,s)=\begin{cases} +k(f(x),1-2s)\ \ \text{if}\ \ s\leq 1/2 \\ +h(x,2s-1)\ \ \ \ \ \ \text{if}\ \ 1/2\leq s.\\ +\end{cases}$$ +The homotopy between homotopies $K$ is specified by +$$K(a,s,t)=\begin{cases} +k(i(a),1-2s(1-t))\ \ \ \ \ \ \ \ \ \ \text{if}\ \ s\leq 1/2 \\ +h(i(a),1-2(1-s)(1-t))\ \ \, \text{if}\ \ s\geq 1/2.\\ +\end{cases}$$ +Traversal of $L$ around the three faces of $I\times I$ other than that specified by $J$ +gives a homotopy +$$e\com f = J_0 = L_{0,0} \htp L_{0,1}\htp L_{1,1}\htp L_{1,0} = J_1=\id \ \text{rel}\ A. \qed$$ +\renewcommand{\qed}{}\end{proof} + +The proposition applies to the following previously encountered situation. + +\begin{exmp} +Let $i: A\rtarr X$ be a cofibration. We then have the commutative diagram +$$\diagram +& A \dlto_j \drto^i \\ +Mi \rrto_r & & X, \\ +\enddiagram$$ +where $j(a)=(a,1)$. The obvious homotopy inverse $\io: X\rtarr Mi$ has $\io(x)=(x,0)$ +and is thus very far from being a map under $A$. The proposition ensures that $\io$ is +homotopic to a map under $A$ that is homotopy inverse to $r$ under $A$. +\end{exmp} + +The following generalization asserts that, for inclusions that are cofibrations, +a pair of homotopy equivalences is a homotopy equivalence of +pairs.\index{homotopy equivalence!of pairs} It is often +used implicitly in setting up homology and cohomology theories on pairs of spaces. + +\begin{prop} Assume given a commutative diagram +$$\diagram +A \rto^{d} \dto_{i} & B \dto^{j} \\ +X \rto_{f} & Y\\ +\enddiagram$$ +in which $i$ and $j$ are cofibrations and $d$ and $f$ are homotopy equivalences. +Then $(f,d):(X,A)\rtarr (Y,B)$ is a homotopy equivalence of pairs. +\end{prop} +\begin{proof} +The statement means that there are homotopy inverses $e$ of $d$ and $g$ +of $f$ such that $g\com j=i\com e$ together with homotopies $H: g\com f\htp\id$ and +$K: f\com g\htp \id$ that extend homotopies $h: e\com d\htp\id$ and $k: d\com e\htp\id$. +Choose any homotopy inverse $e$ to $d$, together with homotopies +$h: e\com d\htp \id$ and $\ell: d\com e\htp\id$. By HEP for $j$, there is a homotopy +inverse $g'$ for $f$ such that $g'\com j = i\com e$. Then, by HEP for $i$, there is +a homotopy $m$ of $g'\com f$ such that $m\com (i\times \id)=i\com h$. Let $\ph = m_1$. Then +$\ph\com i = i$ and $\ph$ is a cofiber homotopy equivalence by the previous result. +Let $\ps: X\rtarr X$ be a homotopy inverse under $i$ and let $n: \ps\com\ph\htp \id$ +be a homotopy under $i$. Define $g= \ps\com g'$. Clearly $g\com j = i\com e$. Using that +the pairs $(I\times I, I\times \sset{0})$ and $(I\times I, I\times \sset{0}\cup \pa I\times I)$ +are homeomorphic, we can construct a homotopy between homotopies $\LA$ by applying HEP to the +diagram +$$\diagram (A\times I \times {0})\cup (A\times \pa I\times I) \ddto_{i\times \id} +\rrto^{\subset} & & A\times I\times I \ddto^{i\times \id} \dlto_{\GA}\\ +& X & \\ +(X\times I \times {0})\cup (X\times \pa I\times I) \urto^{\ga} +\rrto_{\subset} & & X\times I\times I. \ulto_{\LA} \\ +\enddiagram$$ +Here +$$\ga(x,s,0) = \left\{ \begin{array}{ll} + \ps (m(x,2s)) & \mbox{if $s\leq 1/2$} \\ + n(x, 2s-1) & \mbox{if $s\geq 1/2$,} + \end{array} + \right. $$ +$$\ga(x,0,t) = (g\com f)(x) = (\ps\com g'\com f)(x),$$ +and +$$\ga(x,1,t) = x,$$ +while +$$\GA(a,s,t) = \left\{ \begin{array}{ll} + i(h(a,2s/(1+t))) & \mbox{if $2s\leq 1+t$} \\ + i(a) & \mbox{if $2s\geq 1+t$} + \end{array} + \right. $$ +Define $H(x,s)= \LA(x,s,1)$. Then $H: g\com f\htp\id$ and $H\com(i\times\id)=i\com h$. +Application of this argument with $d$ and $f$ replaced by $e$ and $g$ gives a left +homotopy inverse $f'$ to $g$ and a homotopy $L: f'\com g\htp \id$ such that +$f'\com i = j\com d$ and $L\com(j\times \id) = j\com \ell$. Adding homotopies by +concentrating them on successive fractions of the unit interval and letting +the negative of a homotopy be obtained by reversal of direction, define +$$k=(-\ell)(de\times \id) + dh(e\times \id) + \ell$$ +and +$$K=(-L)(fg\times \id) + f'H(g\times \id)+ L.$$ +Then $K: f\com g\htp\id$ and $K\com(j\times \id)= j\com k$. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Show that a cofibration $i: A\rtarr X$ is an inclusion with closed image. +\item Let $i: A\rtarr X$ be a cofibration, where $A$ is a contractible space. +Prove that the quotient map $X\rtarr X/A$ is a homotopy equivalence. +\end{enumerate} + +\chapter{Fibrations} + +We ``dualize'' the definitions and theory of the previous chapter to the study +of fibrations, which are ``up to homotopy'' generalizations of covering spaces. + +\section{The definition of fibrations} + +\begin{defn} +A surjective map $p: E\rtarr B$ is a fibration\index{fibration} if it satisfies the +covering homotopy property (CHP).\index{CHP}\index{covering homotopy property} This +means that if $h\com i_0=p\com f$ in the diagram +$$\diagram +Y \rto^f \dto_{i_0} & E \dto^p \\ +Y\times I \rto_h \urdashed^{\tilde{h}}|>\tip & B,\\ +\enddiagram$$ +then there exists $\tilde{h}$ that makes the diagram commute. +\end{defn} + +This notion of a fibration is due to Hurewicz. There is a more general notion of a +Serre fibration\index{fibration!Serre}, in which the test spaces $Y$ are restricted +to be cubes $I^n$. Serre fibrations are more appropriate for many purposes, but we +shall make no use of them. The test diagram in the definition can be rewritten in the +equivalent form +$$\diagram +E \ddto_{p} & & E^I \llto_{p_0} \ddto^{p^I} \\ +& Y \urdashed^{\tilde{h}}|>\tip \ulto^f \drto_h & \\ +B & & B^I. \llto^{p_0}\\ +\enddiagram$$ +Here $p_0(\be)=\be(0)$ for $\be\in B^I$. With this formulation, we can +``dualize'' the proof that pushouts of cofibrations are cofibrations to +show that pullbacks of fibrations are fibrations. We often write +$A\times_g E$ for the pullback\index{pullback} of a given fibration +$p:E\rtarr B$ and a map $g:A\rtarr B$. + +\begin{lem} +If $p:E\rtarr B$ is a fibration and $g :A\rtarr B$ is any map, then +the induced map $A\times_g E\rtarr A$ is a fibration. +\end{lem} + +\section{Path lifting functions and fibrations} + +Although the CHP is expressed in terms of general test diagrams, there is a +certain universal test diagram. Namely, we can let $Y$ in our original test +diagram be the ``mapping path space''\index{mapping path space} +$$Np \equiv E\times_p B^I =\{ (e,\be)| \be(0)=p(e)\} \subset E\times B^I.$$ +That is, $Np$ is the pullback of $p$ and $p_0$ in the second form of the test +diagram and, with $Y=Np$, $f$ and $h$ in that diagram are the evident projections. +A map $s: Np\rtarr E^I$ +such that $k\com s=\id$, where $k: E^I\rtarr Np$ has coordinates $p_0$ and +$p^I$, is called a path lifting function.\index{path lifting function} Thus +$$s(e,\be)(0)=e \ \ \tand \ \ p\com s(e,\be)=\be.$$ +Given a general test diagram, there results a map $g: Y\rtarr Np$ determined +by $f$ and $h$, and we can take $\tilde{h}= s\com g$. + +In general, path lifting functions are not unique. In fact, we have already +studied the special kinds of fibrations for which they are unique. + +\begin{lem} If $p: E\rtarr B$ is a covering, then $p$ is a fibration with +a unique path lifting function $s$. +\end{lem} +\begin{proof} +The unique lifts of paths with a given initial point specify $s$. +\end{proof} + +Fibrations and cofibrations are related by the following useful observation. + +\begin{lem} +If $i: A\rtarr X$ is a cofibration and $B$ is a space, then the induced map +$$p= B^i: B^X \rtarr B^A$$ +is a fibration. +\end{lem} +\begin{proof} +It is an easy matter to check that we have a homeomorphism +$$B^{Mi} = B^{X\times\sset{0}\cup A\times I} \iso B^X\times_p (B^A)^I = Np.$$ +If $r: X\times I\rtarr Mi$ is a retraction, then +$$B^r: Np \iso B^{Mi}\rtarr B^{X\times I}\iso (B^X)^I$$ +is a path lifting function. +\end{proof} + +\section{Replacing maps by fibrations} + +We can use the mapping path space construction to decompose an arbitrary map +$f: X\rtarr Y$ as the composite of a homotopy equivalence and a fibration. +That is, up to homotopy, any map can be replaced by a fibration. To see this, +recall that $Nf = X\times_f Y^I$ and observe that $f$ +coincides with the composite +$$X \overto{\nu} Nf \overto{\rh} Y,$$ +where $\nu(x)=(x,c_{f(x)})$ and $\rh (x,\ch)=\ch(1)$. Let $\pi: Nf\rtarr X$ +be the projection. Then $\pi\com\nu=\id$ and $\id\htp \nu\com\pi$ since +we can define a deformation $h: Nf\times I\rtarr Nf$ of $Nf$ onto $\nu(X)$ by +setting +$$h(x,\ch)(t)=(x,\ch_t), \ \text{where}\ \ch_t(s)=\ch((1-t)s).$$ +We check directly that $\rh: Nf\rtarr Y$ satisfies the CHP. Consider a test +diagram +$$\diagram +A \rto^g \dto_{i_0} & Nf \dto^{\rh} \\ +A\times I \rto_h \urdashed^{\tilde{h}}|>\tip & Y.\\ +\enddiagram$$ +We are given $g$ and $h$ such that $h\com i_0=\rh\com g$ and must construct +$\tilde{h}$ that makes the diagram commute. We write $g(a)=(g_1(a),g_2(a))$ +and set +$$\tilde{h}(a,t)=(g_1(a),j(a,t)),$$ +where +$$j(a,t)(s)= +\begin{cases} +g_2(a)(s+st) \ \ \ \ \ \text{if}\ \ 0\leq s \leq 1/(1+t)\\ +h(a,s+ts-1) \ \ \text{if} \ \ 1/(1+t) \leq s\leq 1. +\end{cases}$$ + +\section{A criterion for a map to be a fibration} + +Again, we want a criterion that allows us to recognize fibrations when we see +them. Here the idea of duality fails, and we instead think of fibrations as +generalizations of coverings. When restricted to the spaces $U$ in a well +chosen open cover $\sO$ of the base space $B$, a covering is homeomorphic +to the projection $U\times F\rtarr U$, where $F$ is a fixed discrete set. + +The obvious generalization of this is the notion of a bundle. A map $p: E\rtarr B$ +is a bundle\index{bundle} if, when restricted to the spaces $U$ in a well chosen open +cover $\sO$ of $B$, there are homeomorphisms $\ph: U\times F\rtarr p^{-1}(U)$ such +that $p\com \ph= \pi_1$, where $F$ is a fixed topological space. We require of a ``well +chosen'' open cover that it be numerable.\index{numerable open cover} +This means that there are continuous maps $\la_U: B\rtarr I$ such that $\la_U^{-1}(0,1]=U$ +and that the cover is locally finite, in the sense that each $b\in B$ has a neighborhood +that intersects only finitely many $U\in \sO$. Any open cover of a paracompact space has a +numerable refinement. With this proviso on the open covers allowed in the +definition of a bundle, the following result shows in particular that every +bundle is a fibration. + +\begin{thm} Let $p: E\rtarr B$ be a map and let $\sO$ be a numerable open +cover of $B$. Then $p$ is a fibration if and only if $p: p^{-1}(U)\rtarr U$ +is a fibration for every $U\in \sO$. +\end{thm} +\begin{proof} +Since pullbacks of fibrations are fibrations, necessity is obvious. Thus +assume that $p|p^{-1}(U)$ is a fibration for each $U\in \sO$. We shall construct a +path lifting function for $B$ by patching together path lifting functions +for the $p|p^{-1}(U)$, but we first set up the scaffolding of the patching argument. +Choose maps $\la_U: B\rtarr I$ such that $\la_U^{-1}(0,1]=U$. For a finite ordered +subset $T=\sset{U_1,\ldots\!,U_n}$ of sets in $\sO$, +define $c(T)=n$ and define $\la_T: B^I\rtarr I$ by +$$\la_T(\be)= +\text{inf}\{ (\la_{U_i}\com\be)(t)|(i-1)/n\leq t\leq i/n,\ 1\leq i\leq n \}.$$ +Let $W_T=\la_T^{-1}(0,1]$. Equivalently, +$$W_T=\{ \be| \be(t)\in U_i \ \text{if}\ t\in [(i-1)/n,i/n] \} \subset B^I.$$ +The set $\sset{W_T}$ is an open cover of $B^I$, but it need not be locally +finite. However, $\{ W_T|c(T)0\}\subset W_T.$$ +Then $\sset{V_T}$ is a locally finite open cover of $B^I$. We choose a total +ordering of the set of all finite ordered subsets $T$ of $\sO$. + +With this scaffolding in place, choose path lifting functions +$$s_U:p^{-1}(U)\times_pU^I\rtarr p^{-1}(U)^I$$ +for $U\in\sO$, so that $(p\com s_U)(e,\be)=\be$ and $s_U(e,\be)(0)=e$. For a given +$T=\sset{U_1,\ldots\!,U_n}$, consider paths $\be\in V_T$. +For $0\leq u \tip \rto_{\pi_2} & I \rto_{\be} & B.\\ +\enddiagram$$ +At time $t$, $\tilde{\be}$ maps $F_b$ to the fiber $F_{\be(t)}$. In particular, at $t=1$, +this gives a map +$$\ta[\be]\equiv [\tilde{\be_1}]: F_b\rtarr F_{b'},$$ +which we call the translation of fibers along the path class $[\be]$. + +We claim that, as indicated by our choice of notation, the homotopy class of the +map $\tilde{\be_1}$ is independent of the choice of $\be$ in its path class. Thus +suppose that $\be$ and $\be'$ are equivalent paths from $b$ to $b'$, let +$h: I\times I\rtarr B$ be a homotopy $\be\htp\be'$ through paths from $b$ to +$b'$, and let $\tilde{\be}': F_b\times I\rtarr E$ cover $\be'\pi_2$. Observe that if +$$J^2=I\times\pa I\cup \sset{0}\times I\subset I^2,$$ +then the pairs $(I^2,J^2)$ and $(I\times I,I\times\sset{0})$ are homeomorphic. +Define $f: F_b\times J^2 \rtarr E$ to be $\tilde{\be}$ on $F_b\times I\times\sset{0}$, +$\tilde{\be'}$ on $F_b\times I\times\sset{1}$, and $i_b\com \pi_1$ on +$F_b \times\sset{0}\times I$. Then another application of the CHP gives a lift +$\tilde{h}$ in the diagram +$$\diagram +F_b\times J^2 \dto \rrto^f & & E \dto^p \\ +F_b\times I^2 \urrdashed^{\tilde{h}}|>\tip \rto_{\pi_2} & I^2 \rto_{h} & B.\\ +\enddiagram$$ +Thus $\tilde{h}:\tilde{\be}\htp\tilde{\be'}$ through maps $F_b\times I\rtarr E$, +each of which starts at the inclusion of $F_b$ in $E$. At time $t=1$, this gives +a homotopy $\tilde{\be_1}\htp\tilde{\be'_1}$. Thus $\ta[\be] = [\tilde{\be_1}]$ +is a well defined {\em homotopy class} of maps $F_b\rtarr F_{b'}$. + +We think of $\ta[\be]$ as a map in the homotopy category $h\sU$. It is clear +that, in the homotopy category, +$$\ta[c_b]=[\id] \ \ \ \tand \ \ \ \ta[\ga\cdot\be]=\ta[\ga]\com\ta[\be]$$ +if $\ga(0)=\be(1)$. It follows that $\ta[\be]$ is an isomorphism with inverse +$\ta[\be^{-1}]$. This can be stated formally as follows. + +\begin{thm} Lifting of equivalence classes of paths in $B$ to homotopy classes +of maps of fibers specifies a functor $\la: \PI(B)\rtarr h\sU$. Therefore, if +$B$ is path connected, then any two fibers of $B$ are homotopy equivalent. +\end{thm} + +Just as the fundamental group $\pi_1(B,b)$ of the base space of a covering acts +on the fiber $F_b$, so the fundamental group $\pi_1(B,b)$ of the base space of a +fibration acts ``up to homotopy'' on the fiber, in a sense made precise by the +following corollary. For a space $X$, let $\pi_0(X)$ denote the set of path components of $X$. +The set of homotopy equivalences of $X$ is denoted +$\text{Aut}(X)$ and is topologized as a subspace of the function space of maps $X\rtarr X$. +The composite of homotopy equivalences is a homotopy equivalence, and composition defines +a continuous product on $\text{Aut}(X)$. With this product, $\text{Aut}(X)$ is a +``topological monoid,'' namely a space with a continuous and associative multiplication +with a two-sided identity element, but it is not a group. However, the path components +of $\text{Aut}(X)$ are the homotopy classes of homotopy equivalences of $X$, and these +do form a group under composition. + +\begin{cor} +Lifting of equivalence classes of loops specifies a +homomorphism $\pi_1(B,b)\rtarr \pi_0(${\em Aut}$(F_b))$. +\end{cor} + +We have the following naturality statement with respect to maps of fibrations. + +\begin{thm} +Let $p$ and $q$ be fibrations in the commutative diagram +$$\diagram +D \rto^g \dto_q & E \dto^p\\ +A \rto_f & B.\\ +\enddiagram$$ +For a path $\al: I\rtarr A$ from $a$ to $a'$, the following diagram commutes +in $h\sU$: +$$\diagram +F_a \rto^g \dto_{\ta[\al]} & F_{f(a)} \dto^{\ta[f\com\al]} \\ +F_{a'}\rto_g & F_{f(a')}. \\ +\enddiagram$$ +If, further, $h:f\htp f'$ and $H:g\htp g'$ in the commutative diagram +$$\diagram +D\times I \rto^H \dto_{q\times\id} & E \dto^p\\ +A\times I \rto_h & B,\\ +\enddiagram$$ +then the following diagram in $h\sU$ also commutes, where $h(a)(t)=h(a,t)$: +$$\diagram +& F_a \dlto_g \drto^{g'} & \\ +F_{f(a)} \rrto_{\ta[h(a)]} & & F_{f'(a)}. \\ +\enddiagram$$ +\end{thm} +\begin{proof} +Let $\tilde{\al}: F_a\times I\rtarr D$ lift $\al$ and +$\tilde{\be}:F_{f(a)}\times I\rtarr E$ +lift $f\com\al$. Define $j: F_a\times J^2 \rtarr E$ to be $g\com \tilde{\al}$ on +$F_a \times I\times\sset{0}$, $\tilde{\be}\com (g\times\id)$ on +$F_a \times I\times\sset{1}$, and $g\com \pi_1$ on $F_a \times \sset{0}\times I$. Define +$k: I^2\rtarr B$ to be the constant homotopy which at each time $t$ is $f\com \al$. +Another application of the CHP gives a lift $\tilde{k}$ in the diagram: +$$\diagram +F_a\times J^2 \dto \rrto^j & & E \dto^p\\ +F_a\times I^2 \urrdashed^{\tilde{k}}|>\tip \rto_{\pi_2} & I^2 \rto_{k} & B.\\ +\enddiagram$$ +Here $\tilde{k}$ is a homotopy $g\com\tilde{\al}\htp\tilde{\be}\com(g\times\id)$ through +homotopies starting at $g\com\pi_1: F_a\times I\rtarr E$. This gives the diagram +claimed in the first statement. For the second statement, define +$\al: I\rtarr A\times I$ by $\al(t)=(a,t)$, +so that $h(a)=h\com\al$. Define $\tilde{\al}: F_a\rtarr F_a\times I$ by +$\tilde{\al}(f)=(f,t)$. Then $\tilde{\al}$ lifts $\al$ and +$$\ta[\al]=[\id]:F_a=F_a\times \sset{0}\rtarr F_a\times \sset{1}=F_a.$$ +We conclude that the second statement is a special case of the first. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEM +\end{center} +\begin{enumerate} +\item Prove the proposition stated in \S5. +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{Based cofiber and fiber sequences} + +We use cofibrations and fibrations in the category $\sT$ of based spaces +to generate two ``exact sequences of spaces'' from a given map of based +spaces. We shall write $*$ generically for the basepoints of based spaces. +Much that we do for cofibrations can be done equally well in the unbased +context of the previous chapter. However, the dual theory of fibration +sequences only makes sense in the based context. + +\section{Based homotopy classes of maps} + +For based spaces $X$ and $Y$, we let $[X,Y]$ denote the set of based +homotopy classes of based maps $X\rtarr Y$. This set has a natural +basepoint, namely the homotopy class of the constant map from $X$ to +the basepoint of $Y$. + +The appropriate analogue of the Cartesian product in the category of based +spaces is the ``smash product''\index{smash product} $X\sma Y$ defined by +$$ X\sma Y = X\times Y/X\wed Y.$$ +Here $X\wed Y$ is viewed as the subspace of $X\times Y$ consisting of +those pairs $(x,y)$ such that either $x$ is the basepoint of $X$ or +$y$ is the basepoint of $Y$. + +The appropriate based analogue of the function space is the subspace $F(X,Y)$ +of $Y^X$ consisting of the based maps, with the constant based map as +basepoint.\index{function space!based} With these definitions, we have a natural +homeomorphism of based spaces +$$F(X\sma Y,Z)\iso F(X,F(Y,Z))$$ +for based spaces $X$ and $Y$. + +Recall that $\pi_0(X)$ denotes the set of path components of $X$. +When $X$ is based, so is this set, and we sometimes denote it by $\pi_0(X,*)$. +Observe that $[X,Y]$ may be identified with $\pi_0(F(X,Y))$. + +\section{Cones, suspensions, paths, loops} + +Let $X$ be a based space. We define the cone\index{cone} on $X$ to be $CX=X\sma I$, +where $I$ is given the basepoint $1$. That is, +$$CX = X\times I/(\sset{*}\times I\cup X\times\sset{1}).$$ +We view $S^1$ as $I/\pa I$, denote its basepoint by $1$, and define the +suspension\index{suspension} of $X$ to be $\SI X= X\sma S^1$. That is, +$$\SI X= X\times S^1/(\sset{*}\times S^1\cup X\times\sset{1}).$$ +These are sometimes called the reduced cone and suspension,\index{cone!reduced} +\index{suspension!reduced} to distinguish them +from the unreduced constructions, in which the line $\sset{*}\times I$ +through the basepoint of $X$ is not identified to a point. We shall make use of +both constructions in our work, but we shall not distinguish them notationally. + +Dually, we define the path space\index{path space} of $X$ to be $PX=F(I,X)$, where $I$ is +given the basepoint $0$. Thus the points of $PX$ are the paths in $X$ that +start at the basepoint. We define the loop space\index{loop space} of $X$ to be $\OM X=F(S^1,X)$. +Its points are the loops at the basepoint. + +We have the adjunction +$$F(\SI X,Y) \iso F(X,\OM Y).$$ +Passing to $\pi_0$, this gives that +$$[\SI X,Y]\iso [X,\OM Y].$$ + +Composition of loops defines a multiplication on this set. Explicitly, for +$f,g:\SI X\rtarr Y$, we write +$$(g+f)(x\sma t) = (g(x)\cdot f(x))(t)= +\begin{cases} +f(x\sma 2t) \ \ \ \ \ \ \ \ \ \, \text{if} \ 0\leq t\leq 1/2 \\ +g(x\sma(2t-1)) \ \ \text{if} \ 1/2\leq t\leq 1. +\end{cases}$$ + +\begin{lem} $[\SI X,Y]$ is a group and $[\SI^2 X,Y]$ is an Abelian group. +\end{lem} +\begin{proof} +The first statement is proved just as for the fundamental group. For the +second, think of maps $f,g: \SI^2 X\rtarr Y$ as maps $S^2\rtarr F(X,Y)$ +and think of $S^2$ as the quotient $I^2/\pa I^2$. Then a homotopy between +$g+f$ and $f+g$ can be pictured schematically as follows: +$$\diagram +\rrline \ddline & & \ddline & +\rrline \ddline & \ddline & \ddline & +\rrline \ddline & \ddline & \ddline & +\rrline \ddline & & \ddline \\ +\rrline^{^f} & & \rto & \rrline^<(0.25){^*} ^<(0.75){^f} & & \rto +& \rrline^<(0.25){^g} ^<(0.75){^*} & & \rto & \rrline^{^g} & & \\ +\rrline^{^g} & & & \rrline^<(0.25){^g} ^<(0.75){^*} & & +& \rrline^<(0.25){^*} ^<(0.75){^f} & & & \rrline^{^f} & & \\ +\enddiagram$$ +\end{proof} + +\section{Based cofibrations} + +The definition of a cofibration\index{cofibration!based} has an evident based variant, in which +all given and constructed maps in our test diagrams are required to be +based. A based map $i: A\rtarr X$ that is a cofibration in the unbased +sense is necessarily a cofibration in the based sense since the basepoint +of $X$ must lie in $A$. + +We say that $X$ is ``nondegenerately based,'' or ``well pointed,''\index{nondegenerately +based space}\index{well pointed space} if the inclusion of its basepoint is a cofibration in the unbased +sense. If $A$ and $X$ are nondegenerately based and $i: A\rtarr X$ is a based cofibration, +then $i$ is necessarily an unbased cofibration. + +We refer to based cofibrations simply as cofibrations in the rest of this +chapter. + +Write $Y_+$ for the union of a space $Y$ and a disjoint basepoint and observe +that we can identify $X\sma Y_+$ with $X\times Y/\sset{*}\times Y$. + +The space $X\sma I_+$ is called the reduced cylinder\index{cylinder!reduced} on $X$, and a based +homotopy\index{homotopy!based} $X\times I\rtarr Y$ is the same thing as a based map +$X\sma I_+\rtarr Y$. We change notations and write $Mf$ for the based +mapping cylinder $Y\cup_f (X\sma I_+)$ of a based map $f$. + +As in the unbased case, we conclude that a based map $i:A\rtarr X$ is a +cofibration if and only if $Mi$ is a retract of $X\sma I_+$. + +\section{Cofiber sequences} + +For a based map $f:X\rtarr Y$, define the ``homotopy cofiber''\index{homotopy cofiber} $Cf$ to be +$$Cf=Y\cup_f CX= Mf/j(X),$$ +where $j: X\rtarr Mf$ sends $x$ to $(x,1)$. As in the unbased case, our +original map $f$ is the composite of the cofibration $j$ and the evident +retraction $r: Mf\rtarr Y$. Thus $Cf$ is constructed by first replacing +$f$ by the cofibration $j$ and then taking the associated quotient space. + +Let $i: Y\rtarr Cf$ be the inclusion. It is a cofibration since it is the +pushout of $f$ and the cofibration $X\rtarr CX$ that sends $x$ to $(x,0)$. +Let +$$\pi:Cf\rtarr Cf/Y\iso \SI X$$ +be the quotient map. The sequence +$$ X\overto{f} Y\overto{i} Cf\overto{\pi} \SI X +\overto{-\SI f} \SI Y \overto{-\SI i} \SI Cf \overto{-\SI \pi} \SI^2 X +\overto{\SI^2f} \SI^2 Y\rtarr\cdots$$ +is called the cofiber sequence\index{cofiber sequence} generated by the map $f$; here +$$(-\SI f)(x\sma t)=f(x)\sma(1-t).$$ + +These ``long exact sequences of based spaces''\index{long exact sequence!of based spaces} +give rise to long exact +sequences of pointed sets,\index{long exact sequence!of pointed sets} where a sequence +$$S'\overto{f} S\overto{g} S''$$ +of pointed sets is said to be exact if $g(s)=*$ if and only if $s=f(s')$ +for some $s'$. + +\begin{thm} +For any based space $Z$, the induced sequence +$$\cdots \rtarr [\SI Cf,Z] \rtarr [\SI Y,Z]\rtarr [\SI X,Z] +\rtarr [Cf,Z]\rtarr [Y,Z]\rtarr [X,Z]$$ +is an exact sequence of pointed sets, or of groups to the left of $[\SI X,Z]$, +or of Abelian groups to the left of $[\SI^2 X,Z]$. +\end{thm} + +Exactness is clear at the first stage, where we are considering the composite of +$f: X\rtarr Y$ and the inclusion $i$ of $Y$ in the cofiber $Cf$. To see this, consider +the diagram +$$\diagram +X\rto^f & Y \dto_g \rto^(0.2)i & Cf= Y\cup_f CX \dldashed^{\tilde{g}=g\cup h}|>\tip \\ +& Z. &\\ +\enddiagram$$ +Here $h:g\com f\htp c_*$, and we view $h$ as a map $CX\rtarr Z$. Thus we +check exactness by using any given homotopy to extend $g$ over the cofiber. We emphasize +that this applies to any composite pair of maps of the form $(f,i)$, where $i$ is the +inclusion of the target of $f$ in the cofiber of $f$. + +We claim that, up to homotopy equivalence, each consecutive pair of maps +in our cofiber sequence is the composite of a map and the inclusion of its +target in its cofiber. This will imply the theorem. We observe that, for any +map $f$, interchange of the cone and suspension coordinate gives a homeomorphism +$$\SI Cf\iso C(\SI f)$$ +such that the following diagram commutes: +$$\diagram +\SI X\rto^{\SI f} \ddouble & \SI Y \ddouble \rto^{\SI i(f)} & \SI Cf \rto^{\SI \pi(f)} \dto^{\iso} +& \SI^2X\dto^{\tau} \\ +\SI X\rto_{\SI f} & \SI Y \rto_(0.4){ i(\SI f)} & C (\SI f) \rto_{\pi(\SI f)} & \SI^2X.\\ +\enddiagram$$ +Here $\ta: \SI^2X\rtarr \SI^2X$ is the homeomorphism obtained by interchanging the two +suspension coordinates; we shall see later, and leave as an exercise here, that $\ta$ is +homotopic to $-\id$. We have written $i(f)$, $\pi(f)$, etc., to indicate the maps to which the +generic constructions $i$ and $\pi$ are applied. Using this inductively, we see that we need only +verify our claim for the two pairs of maps $(i(f),\pi(f))$ and $(\pi(f),-\SI f)$. The +following two lemmas will imply the claim in these two cases. More precisely, they will +imply the claim directly for the first pair and will imply that the second pair is equivalent +to a pair of the same form as the first pair. + +\begin{lem} +If $i: A\rtarr X$ is a cofibration, then the quotient map +$$\ps: Ci\rtarr Ci/CA\iso X/A$$ +is a based homotopy equivalence. +\end{lem} +\begin{proof} +Since $i$ is a cofibration, there is a retraction +$$r: X\sma I_+\rtarr Mi = X\cup_i (A\sma I_+).$$ +We embed $X$ as $X\times\sset{1}$ in the source and collapse out +$A\times\sset{1}$ from the target. The resulting composite $X\rtarr Ci$ +maps $A$ to $\sset{*}$ and so induces a map $\ph: X/A\rtarr Ci$. The +map $r$ restricts to the identity on $A\sma I_+$, and if we collapse out +$A\sma I_+$ from its source and target, then $r$ becomes a homotopy +$\id\htp \ps\com\ph$. The map $r$ on $X\sma I_+$ glues together with the +map $h: CA\sma I_+\rtarr CA$ specified by +$$h(a,s,t)=(a,\text{max}(s,t))$$ +to give a homotopy $Ci\sma I_+\rtarr Ci$ from the identity to $\phi\com\ps$. +\end{proof} + +\begin{lem} +The left triangle commutes and the right triangle commutes up to homotopy +in the diagram +$$\diagram + X\rto^{f} & Y\rto^{i(f)} & Cf\rto^{\pi(f)} \drto_{i(i(f))} & \SI X +\rto^{-\SI f} & \SI Y \rto & \cdots \\ + & & & Ci(f) \uto_{\ps} \urto_{\pi(i(f))}& & \\ +\enddiagram$$ +\end{lem} +\begin{proof} +Observe that $Ci(f)$ is +obtained by gluing the cones $CX$ and $CY$ along their bases via the map +$f: X\rtarr Y$. The left triangle commutes since collapsing out $CY$ from +$Ci(f)$ is the same as collapsing out $Y$ from $Cf$. A homotopy +$h: Ci(f)\sma I_+\rtarr \SI Y$ from $\pi$ to $(-\SI f)\com\ps$ is given by +$$h(x,s,t)=(f(x),t-st) \ \ \text{on} \ \ CX$$ +and +$$h(y,s,t)=(y,s+t-st) \ \ \ \text{on} \ \ CY. \qed$$ +\renewcommand{\qed}{}\end{proof} + +\section{Based fibrations} + +Similarly, the definition of a fibration\index{fibration!based} has an evident based variant, +in which all given and constructed maps in our test diagrams are required +to be based. A based fibration $p: E\rtarr B$ is necessarily a fibration +in the unbased sense, as we see by restricting to spaces of the form $Y_+$ +in test diagrams and noting that $Y_+\sma I_+\iso (Y\times I)_+$. Less +obviously, if $p$ is a based map that is an unbased fibration, then it +satisfies the based CHP for test diagrams in which $Y$ is nondegenerately +based. + +We refer to based fibrations simply as fibrations in the rest of this +chapter. + +Observe that a based homotopy\index{homotopy!based} $X\sma I_+\rtarr Y$ is the same thing as a +based map $X\rtarr F(I_+,Y)$. Here $F(I_+,Y)$ is the same space as $Y^I$, +but given a basepoint determined by the basepoint of $Y$. Therefore the based version +of the mapping path space $Nf$\index{mapping path space} of a based map $f:X\rtarr Y$ +is the same space +as the unbased version, but given a basepoint determined by the given basepoints +of $X$ and $Y$. However, because path spaces are always defined with $I$ having +basepoint $0$ rather than $1$, we find it convenient to redefine +$Nf$ correspondingly, setting +$$Nf=\{ (x,\ch)|\ch (1)=f(x)\} \subset X\times Y^I.$$ + +As in the unbased case, we easily check that a based map $p:E\rtarr B$ is a +fibration if and only if there is a based path lifting function\index{path lifting function!based} +$$s: Np\rtarr F(I_+,E).$$ + +\section{Fiber sequences} + +For a based map $f:X\rtarr Y$, define the ``homotopy fiber''\index{homotopy fiber} $Ff$ to be +$$Ff=X\times_f PY= \{ (x,\ch)| f(x)=\ch(1)\}\subset X\times PY.$$ +Equivalently, $Ff$ is the pullback displayed in the diagram +$$\diagram +Ff\rto \dto_{\pi} & PY \dto^{p_1}\\ +X\rto_f & Y, \\ +\enddiagram$$ +where $\pi(x,\ch)=x$. As a pullback of a fibration, $\pi$ is a fibration. + +If $\rh: Nf\rtarr Y$ is defined by $\rh(x,\ch)=\ch(0)$, then $f=\rh\com\nu$, +where $\nu(x)=(x,c_{f(x)})$, and $Ff$ is the fiber $\rh^{-1}(*)$. Thus the +homotopy fiber $Ff$ is constructed by first replacing $f$ by the fibration +$\rh$ and then taking the actual fiber. + +Let $\io: \OM Y\rtarr Ff$ be the inclusion specified by $\io(\ch)=(*,\ch)$. +The sequence + +$$ \cdots \rtarr \OM^2X \overto{\OM^2f} \OM^2Y \overto{-\OM\io} \OM Ff +\overto{-\OM \pi} \OM X +\overto{-\OM f} \OM Y \overto{\io} Ff \overto{\pi} X +\overto{f} Y$$ +is called the fiber sequence generated by the map $f$; here +$$(-\OM f)(\ze)(t)=(f\com \ze)(1-t) \ \ \text{for}\ \ze\in \OM X.$$ + +These ``long exact sequences of based spaces''\index{long exact sequence!of based spaces} +also give rise to long exact +sequences of pointed sets,\index{long exact sequence!of pointed sets} this time covariantly. + +\begin{thm} +For any based space $Z$, the induced sequence +$$\cdots \rtarr [Z,\OM Ff]\rtarr [Z,\OM X] \rtarr [Z,\OM Y]\rtarr [Z,Ff] +\rtarr [Z,X]\rtarr [Z,Y]$$ +is an exact sequence of pointed sets, or of groups to the left of $[Z,\OM Y]$, +or of Abelian groups to the left of $[Z,\OM^2 Y]$. +\end{thm} + +Exactness is clear at the first stage. To see this, consider the diagram +$$\diagram +& Z \dldashed_{\tilde{g}=g\times h}|>\tip \dto^g &\\ +Ff=X\times_f PY\rto_(0.7){\pi} & X \rto_f & Y\\ +\enddiagram$$ +Here $h:c_*\htp f\com g$, and we view $h$ as a map $Z\rtarr PY$. Thus we +check exactness by using any given homotopy to lift $g$ to the fiber. + +We claim that, up to homotopy equivalence, each consecutive pair of maps +in our fiber sequence is the composite of a map and the projection from its +fiber onto its source. This will imply the theorem. We observe that, for any +map $f$, interchange of coordinates gives a homeomorphism +$$\OM Ff\iso F(\OM f)$$ +such that the following diagram commutes: +$$\diagram +\OM^2 Y\rto^{\OM \io(f)} \dto_{\ta} & \OM Ff \dto^{\iso}\rto^(0.55){\OM \pi(f)} & \OM X \rto^{\OM f} +\ddouble +& \OM Y\ddouble \\ +\OM^2 Y\rto_(0.45){\io(\OM f)} & F(\OM f) \rto_(0.55){ \pi(\OM f)} & \OM X \rto_{\OM f} & \OM Y.\\ +\enddiagram$$ +Here $\ta$ is obtained by interchanging the loop coordinates and is homotopic to $-\id$. +We have written $\io(f)$, $\pi(f)$, etc., to indicate the maps to which the generic +constructions $\io$ and $\pi$ are applied. Using this inductively, we see that we need only +verify our claim for the two pairs of maps $(\io(f),\pi(f))$ and $(-\OM f,\io(f))$. The +following two lemmas will imply the claim in these two cases. More precisely, they will +imply the claim directly for the first pair and will imply that the second pair is equivalent +to a pair of the same form as the first pair. The proofs of the lemmas are left as exercises. + +\begin{lem} +If $p: E\rtarr B$ is a fibration, then the inclusion +$$\ph: p^{-1}(*)\rtarr Fp$$ +specified by $\ph (e)=(e,c_*)$ is a based homotopy equivalence. +\end{lem} + +\begin{lem} +The right triangle commutes and the left triangle commutes up to homotopy +in the diagram +$$\diagram +\cdots \rto & \OM X \rto^{-\OM f} \drto_{\io(\pi (f))} & \OM Y +\dto^{\ph} \rto^{\io (f)} & Ff \rto^{\pi (f)} & X \rto^f & Y.\\ + & & F\pi (f) \urto_{\pi (\pi (f))} & & & \\ +\enddiagram$$ +\end{lem} + +\section{Connections between cofiber and fiber sequences} + +It is often useful to know that cofiber sequences and fiber sequences can be +connected to one another. The adjunction between loops and suspension has +``unit'' and ``counit'' maps +$$\et: X\rtarr \OM\SI X\ \ \tand \ \ \epz: \SI\OM X\rtarr X.$$ +Explicitly, $\et(x)(t)=x\sma t$ and $\epz(\ch\sma t)= \ch(t)$ for $x\in X$, $\ch\in\OM X$, +and $t\in S^1$. For a map $f: X\rtarr Y$, we define +$$\et: Ff\rtarr \OM Cf \ \ \tand \ \ \epz: \SI Ff\rtarr Cf$$ +by +$$\et(x,\ga)(t)=\epz(x,\ga,t)=\begin{cases} +\ga(2t) \ \ \ \ \ \ \ \ \ \ \text{if}\ \ t\leq 1/2 \\ +(x,2t-1) \ \ \ \, \text{if}\ \ t\geq 1/2\\ +\end{cases}$$ +for $x\in X$ and $\ga\in PY$ such that $\ga(1)=f(x)$. Thus $\epz$ is just the +adjoint of $\et$. + +\begin{lem} Let $f: X\rtarr Y$ be a map of based spaces. Then the following +diagram, in which the top row is the suspension of part of the fiber sequence +of $f$ and the bottom row is the loops on part of the cofiber sequence of $f$, +is homotopy commutative: +$$\diagram + & \SI\OM Ff \dto_{\epz} \rto^{\SI\OM p} & \SI\OM X \dto_{\epz} \rto^{\SI\OM f} +& \SI \OM Y \rto^{\SI\io} \dto^{\epz} +& \SI Ff \dto^{\epz} \rto^{\SI p} & \SI X \ddouble \\ +\OM Y\rto^{\io} \ddouble & Ff \rto^{p} \dto_{\eta} & X \rto^{f} \dto_{\et} & +Y \rto^{i} \dto^{\et} & Cf \rto^{\pi} \dto^{\et} & \SI X \\ +\OM Y\rto_{\OM i} & \OM Cf \rto_{\OM\pi} & +\OM\SI X \rto_{\OM\SI f} & \OM\SI Y \rto_{\OM\SI i} & \OM\SI Cf.\\ +\enddiagram$$ +\end{lem} +\begin{proof} +Four of the squares commute by naturality and the remaining four squares +consist of two pairs that are adjoint to each other. To see that the +two bottom left squares commute up to homotopy one need only write down +the relevant maps explicitly. +\end{proof} + +Another easily verified result along the same lines relates the quotient +map $(Mf,X)\rtarr (Cf,*)$ to $\et: Ff\rtarr \OM Cf$. Here in the based +context we let $Mf$ be the reduced mapping cylinder, in which the line +through the basepoint of $X$ is collapsed to a point. + +\begin{lem} +Let $f: X\rtarr Y$ be a map of based spaces. Then the following diagram is +homotopy commutative, where $j: X\rtarr Mf$ is the inclusion, $r:Mf\rtarr Y$ +is the retraction, and $\pi$ is induced by the quotient map $Mf\rtarr Cf$: +$$\diagram +Fj=X\times_jPMf\rrto^{Fr=\id\times Pr} \drto_{\pi} & &X\times_f PY=Ff \dlto^{\et}\\ +& \OM Cf. & \\ +\enddiagram$$ +\end{lem} + +\vspace{.1in} + +\begin{center} +PROBLEM +\end{center} +\begin{enumerate} +\item Prove the two lemmas stated at the end of \S6. +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{Higher homotopy groups} + +The most basic invariants in algebraic topology are the homotopy groups. +They are very easy to define, but very hard to compute. We give the +basic properties of these groups here. + +\section{The definition of homotopy groups} + +For $n\geq 0$ and a based space $X$, define\index{homotopy groups} +$$\pi_n(X)=\pi_n(X,*)=[S^n,X],$$ +the set of homotopy classes of based maps $S^n\rtarr X$. This is a +group if $n\geq 1$ and an Abelian group if $n\geq 2$. When $n=0$ and +$n=1$, this agrees with our previous definitions. Observe that +$$\pi_n(X)=\pi_{n-1}(\OM X)=\cdots=\pi_0(\OM^nX).$$ + +For $*\in A\subset X$, the (homotopy) fiber of the inclusion $A\rtarr X$ +may be identified with the space $P(X;*,A)$ of paths in $X$ that begin at +the basepoint and end in $A$. For $n\geq 1$, define +$$\pi_n(X,A)=\pi_n(X,A,*)=\pi_{n-1}P(X;*,A).$$ +This is a group if $n\geq 2$ and an Abelian group if $n\geq 3$. Again, +$$\pi_n(X,A)=\pi_0(\OM^{n-1}P(X;*,A)).$$ +These are called relative homotopy groups.\index{homotopy groups!relative} + +\section{Long exact sequences associated to pairs} + +With $Fi=P(X;*,A)$, we have the fiber sequence +$$\cdots\rtarr \OM^2A\rtarr \OM^2X \rtarr \OM Fi\rtarr \OM A +\rtarr \OM X \overto{\io} Fi\overto{p_1} A \overto{i}X$$ +associated to the inclusion $i: A\rtarr X$, where $p_1$ is the endpoint +projection and $\io$ is the inclusion. Applying the functor +$\pi_0(-)=[S^0,-]$ to this sequence, we obtain the long exact sequence +$$\cdots\rtarr \pi_n(A) \rtarr \pi_n(X) \rtarr \pi_n(X,A) +\overto{\pa} \pi_{n-1}(A) +\rtarr \cdots \rtarr \pi_0(A)\rtarr \pi_0(X).$$ + +Define +$$J^n=\pa I^{n-1}\times I\cup I^{n-1}\times\sset{0}\subset I^n,$$ +with $J^1=\sset{0}\subset I$. We can write +$$\pi_n(X,A,*)=[(I^n,\pa I^n,J^n),(X,A,*)],$$ +where the notation indicates the homotopy classes of maps of triples:\index{triple} +maps and homotopies carry $\pa I^n$ into A and $J^n$ to the basepoint. Then +$$\pa: \pi_n(X,A)\rtarr \pi_{n-1}(A)$$ +is obtained by restricting maps +$$(I^n,\pa I^n,J^n)\rtarr (X,A,*)$$ +to maps +$$(I^{n-1}\times\sset{1},\pa I^{n-1}\times\sset{1})\rtarr (A,*),$$ +while $\pi_n(A)\rtarr \pi_n(X)$ and $\pi_n(X)\rtarr \pi_n(X,A)$ are +induced by the inclusions +$$(A,*)\subset (X,*) \ \ \ \tand \ \ \ (X,*,*)\subset (X,A,*).$$ + +\section{Long exact sequences associated to fibrations} + +Let $p: E\rtarr B$ be a fibration, where $B$ is path connected. Fix a +basepoint $*\in B$, let $F=p^{-1}(*)$, and fix a basepoint $*\in F\subset E$. +The inclusion $\ph: F\rtarr Fp$ is a homotopy equivalence, and, being +pedantically careful to choose signs appropriately, we obtain the following +diagram, in which two out of each three consecutive squares commute and the +third commutes up to homotopy: +$$\diagram +\cdots\rto & \OM^2E \rto^{-\OM\io} \dto^{\id} +& \OM Fi \rto^{-\OM p_1} \dto^{-\OM p} & \OM F \dto^{\OM\ph} +\rto^{-\OM i} & \OM E \dto^{\id} \rto^{\io} & Fi \rto^{p_1} \dto^{-p} +& F \rto^{i} \dto^{\ph} & E \dto^{\id} \\ +\cdots\rto & \OM^2E \rto_{\OM^2p} & \OM^2B \rto_{-\OM \io} & \OM Fp +\rto_{-\OM \pi} & \OM E \rto_{-\OM p} & \OM B\rto_{\io} & Fp \rto_{\pi} + & E.\\ +\enddiagram$$ +Here $Fi=P(E;*,F)$, $p(\xi)=p\com \xi\in\OM B$ for $\xi\in Fi$, and the next to +last square commutes up to the homotopy $h: \io\com(-p)\htp\ph\com p_1$ +specified by +$$h(\xi,t)=(\xi(t),p(\xi[1,t])),$$ +where $\xi[1,t](s)=\xi(1-s+st)$. + +Passing to long exact sequences of homotopy groups and using the five lemma, +together with a little extra argument in the case $n=1$, we conclude that +$$p_*: \pi_n(E,F)\rtarr \pi_n(B)$$ +is an isomorphism for $n\geq 1$. This can also be derived directly from the +covering homotopy property. + +Using $\ph_*$ to identify $\pi_*F$ with $\pi_*(Fp)$, we may rewrite the long +exact sequence of the bottom row of the diagram as +$$\cdots\rtarr \pi_n(F) \rtarr \pi_n(E) \rtarr \pi_n(B) +\overto{\pa} \pi_{n-1}(F) +\rtarr \cdots \rtarr \pi_0(E)\rtarr \sset{*}.$$ +(At the end, a little path lifting argument shows that $\pi_0(F)\rtarr \pi_0(E)$ is +a surjection.) This is one of the main tools for the computation of homotopy groups. + +\section{A few calculations} + +We observe some easily derived calculational facts about homotopy groups. + +\begin{lem} If $X$ is contractible, then $\pi_n(X)=0$ for all $n\geq 0$. +\end{lem} + +\begin{lem} If $X$ is discrete, then $\pi_n(X)=0$ for all $n > 0$. +\end{lem} + +\begin{lem} If $p: E\rtarr B$ is a covering, then +$p_*: \pi_n(E)\rtarr \pi_n(B)$ is an isomorphism for all $n\geq 2.$ +\end{lem} + +\begin{lem} $\pi_1(S^1)=\bZ$ and $\pi_n(S^1)=0$ if $n\neq 1$. +\end{lem} + +\begin{lem} If $i\geq 2$, then $\pi_1(\bR P^i)=\bZ_2$ and +$\pi_n(\bR P^i)\iso \pi_n(S^i)$ for $n\neq 1$. +\end{lem} + +\begin{lem} For all spaces $X$ and $Y$ and all $n$, +$$\pi_n(X\times Y)\iso \pi_n(X)\times \pi_n(Y).$$ +\end{lem} + +\begin{lem} If $in>1$ are all finite +except for $\pi_{4n-1}(S^{2n})$, which is the direct sum of $\bZ$ and +a finite group. + +The difficulty of computing homotopy groups is well illustrated by the fact +that there is no non-contractible simply connected compact manifold (or +finite CW complex) all of whose homotopy groups are known. We shall find +many non-compact spaces whose homotopy groups we can determine completely. +Such computations will rely on the following observation. + +\begin{lem} +If $X$ is the colimit of a sequence of inclusions $X_i\rtarr X_{i+1}$ +of based spaces, then the natural map +$$ \colim_i\pi_n(X_i)\rtarr \pi_n(X)$$ +is an isomorphism for each $n$. +\end{lem} +\begin{proof} +This follows directly from the point-set topological fact that if $K$ is a +compact space, then a map $K\rtarr X$ has image in one of the $X_i$. +\end{proof} + +\section{Change of basepoint} + +We shall use our results on change of fibers to generalize our results on +change of basepoint from the fundamental group to the higher absolute and +relative homotopy groups. In the absolute case, we have the identification +$$\pi_n(X,x)=[(S^n,*),(X,x)],$$ +where we assume that $n\geq 1$. Since the inclusion of the basepoint in +$S^n$ is a cofibration, evaluation +at the basepoint gives a fibration $p: X^{S^n}\rtarr X$. We may identify +$\pi_n(X,x)$ with $\pi_0(F_x)$ since a path in $F_x$ is just a based homotopy +$h: S^n\times I\rtarr X$ with respect to the basepoint $x$. Another way to see +this is to observe that $F_x$ is the $n$th loop space $\OM^nX$, specified with +respect to the basepoint $x$. A path class +$[\xi]: I\rtarr X$ from $x$ to $x'$ induces a homotopy equivalence +$\ta[\xi]: F_x\rtarr F_{x'}$, and we continue to write $\ta[\xi]$ for +the induced bijection +$$\ta[\xi]: \pi_n(X,x)\rtarr \pi_n(X,x').$$ + +This bijection is an isomorphism of groups. One conceptual way to see this +is to observe that addition is induced from the ``pinch map'' $S^n\rtarr S^n\wed S^n$ +that is obtained by collapsing an equator to the basepoint. That is, the sum of maps +$f,g: S^n\rtarr X$ is the composite +$$S^n \rtarr S^n\wed S^n \overto{f\wed g} X\wed X\overto{\triangledown} X,$$ +where $\triangledown$ is the folding map, which restricts to the identity map +$X\rtarr X$ on each wedge summand. Evaluation at the basepoint of $S^n\wed S^n$ gives +a fibration $X^{S^n\wed S^n}\rtarr X$, and the pinch map induces a map of fibrations +$$\diagram +X^{S^n\wed S^n} \dto \rto & X^{S^n} \dto\\ +X \rdouble & X.\\ +\enddiagram$$ +The fiber over $x$ in the left-hand fibration is the product $F_x\times F_x$, +where $F_x$ is the fiber over $x$ in the right-hand fibration. In fact, the induced map of +fibers can be identified as the map $\OM^nX\times \OM^nX\rtarr \OM^nX$ given by composition +of loops (using the first loop coordinate say). By the naturality of +translations of fibers with respect to maps of fibrations, we have a homotopy commutative +diagram +$$\diagram +F_x\times F_x \rto \dto_{\ta[\xi]\times \ta[\xi]} & F_x \dto^{\ta[\xi]}\\ +F_{x'}\times F_{x'} \rto & F_{x'} +\enddiagram$$ +in which the horizontal arrows induce addition on passage to $\pi_0$. + +We can argue similarly in the relative case. The triple $(I^n,\pa I^n,J^n)$ +is homotopy equivalent to the triple $(CS^{n-1},S^{n-1},*)$, as we see by +quotienting out $J^n$. Therefore, for $a\in A$, we have the identification +$$\pi_n(X,A,a)\iso [(CS^{n-1},S^{n-1},*),(X,A,a)].$$ +Using that the inclusions $\sset{*}\rtarr S^{n-1}$ and $S^{n-1}\rtarr CS^{n-1}$ +are both cofibrations, we can check that evaluation at $*$ specifies a fibration +$$p: (X,A)^{(CS^{n-1},S^{n-1})}\rtarr A,$$ +where the domain is the subspace of $X^{CS^{n-1}}$ consisting of the +indicated maps of pairs. We may identify $\pi_n(X,A,a)$ with $\pi_0(F_a)$. A path +class $[\al]: I\rtarr A$ from $a$ to $a'$ induces a homotopy equivalence +$\ta[\al]: F_a\rtarr F_{a'}$, and we continue to write $\ta[\al]$ for +the induced isomorphism +$$\ta[\al]: \pi_n(X,A,a)\rtarr \pi_n(X,A,a').$$ +Our naturality results on change of fibers now directly imply the desired results +on change of basepoint. + +\begin{thm} +If $f: (X,A)\rtarr (Y,B)$ is a map of pairs and $\al: I\rtarr A$ is a path +from $a$ to $a'$, then the following diagram commutes: +$$\diagram +\pi_n(X,A,a) \dto_{\ta[\al]} \rto^{f_*} & \pi_n(Y,B,f(a)) \dto^{\ta[f\com\al]}\\ +\pi_n(X,A,a') \rto^{f_*} & \pi_n(Y,B,f(a'))\\ +\enddiagram$$ +If $h:f\htp f'$ is a homotopy of maps of pairs and $h(a)(t)=h(a,t)$, then the +following diagram commutes: +$$\diagram +& \pi_n(X,A,a) \dlto_{f_*} \drto^{f'_*} &\\ +\pi_n(Y,B,f(a)) \rrto_{\ta[h(a)]} & & \pi_n(Y,B,f'(a)).\\ +\enddiagram$$ +The analogous conclusions hold for the absolute homotopy groups. +\end{thm} + +Therefore, up to non-canonical isomorphism, the homotopy groups of $(X,A)$ are +independent of the choice of basepoint in a given path component of $A$. + +\begin{cor} +A homotopy equivalence of spaces or of pairs of spaces induces an isomorphism +on all homotopy groups. +\end{cor} + +We shall soon show that the converse holds for a quite general class of spaces, +namely the class of CW complexes, but we first need a few preliminaries. + +\section{$n$-Equivalences, weak equivalences, and a technical lemma} + +\begin{defn} +A map $e: Y\rtarr Z$ is an $n$-equivalence\index{nequivalence@$n$-equivalence} if, for all $y\in Y$, the map +$$e_*:\pi_q(Y,y)\rtarr \pi_q(Z,e(y))$$ +is an injection for $q\tip & & +CS^n \uldashed^{\tilde{g}}|>\tip \llto^{i_1} \\ +\enddiagram $$ +\item[(iii)] The conclusion of (ii) holds when $f|S^n=e\com g$ and $h$ is the +constant homotopy at this map. +\end{enumerate} +\end{lem} +\begin{proof} +Trivially (ii) implies (iii). We first show that (iii) implies (i). If $n=0$, (iii) +says (in part) that if $e(y)$ and $e(y')$ can be connected by a path in $Z$, then +$y$ and $y'$ can be connected by a path in $Y$. If $n>0$, then (iii) says (in part) +that if $e\com g$ is null homotopic, then $g$ is null homotopic. Therefore +$\pi_n(e)$ is injective. If we specialize (iii) by letting $g$ be the constant map +at a point $y\in Y$, then $f$ is a map $(CS^n,S^n)\rtarr (Z,e(y))$, $\tilde{g}$ is a map +$(CS^n,S^n)\rtarr (Y,y)$, and $\tilde{h}: f\htp e\com \tilde{g}\ \text{rel}\ S^n$. +Therefore $\pi_{n+1}(e)$ is surjective. + +Thus assume (i). We must prove (ii), and we assume given $f$, $g$, and $h$ making +the solid arrow part of the diagram commute. The idea is to use (i) to show that +the $n$th homotopy group of the fiber $F(e)$ is zero, to use the given part of the +diagram to construct a map $S^n\rtarr F(e)$, and to use a null homotopy of that +map to construct $\tilde{g}$ and $\tilde{h}$. However, since homotopy groups involve +choices of basepoints and the diagram makes no reference to basepoints, the details +require careful tracking of basepoints. Thus fix a basepoint $*\in S^n$, let $\bullet$ be +the cone point of $CS^n$, and define +$$y_1=g(*),\ \ z_1=e(y_1),\ \ z_0=f(*,0),\ \ \tand\ \ z_{-1}=f(\bullet).$$ +For $x\in S^n$, let $f_x: I\rtarr Z$ and $h_x: I\rtarr Z$ be the paths +$f_x(s)=f(x,s)$ from $f(x,0)=h(x,0)$ to $z_{-1}$ and $h_x(t)=h(x,t)$ from $h(x,0)$ to +$h(x,1)=(e\com g)(x)$. Consider the homotopy fiber +$$F(e;y_1)=\sset{(y,\ze)|\ze(0)=z_1 \, \tand \, e(y)=\ze(1)}\subset Y\times Z^I.$$ +This has basepoint $w_1=(y_1,c_{z_1})$. By (i) and the exact sequence +$$\pi_{n+1}(Y,y_1)\overto{e_*}\pi_{n+1}(Z,z_1)\rtarr \pi_n(F(e;y_1),w_1)\rtarr +\pi_n(Y,y_1)\overto{e_*}\pi_n(Z,z_1),$$ +we see that $\pi_n(F(e;y_1),w_1)=0$. Define $k_0: S^n\rtarr F(e;y_1)$ by +$$k_0(x)=(g(x),h_x\cdot f_x^{-1}\cdot f_*\cdot h_*^{-1}).$$ +While $k_0$ is not a based map, $k_0(*)$ is connected to the basepoint since +$h_*\cdot f_*^{-1}\cdot f_*\cdot h_*^{-1}$ is equivalent to $c_{z_1}$. By HEP for +the cofibration $\sset{*}\rtarr S^n$, $k_0$ is homotopic to a based map. This based map is +null homotopic in the based sense, hence $k_0$ is null homotopic in the unbased +sense. Let $k: S^n\times I\rtarr F(e;y_1)$ be a homotopy from $k_0$ to the +trivial map at $w_1$. Write +$$k(x,t)=(\tilde{g}(x,t),\ze(x,t)).$$ +Then $\tilde{g}(x,1)=y_1$ for all $x\in S^n$, so that $\tilde{g}$ factors through +a map $CS^n\rtarr Y$, and $\tilde{g}=g$ on $S^n$. We have a map +$j: S^n\times I\times I$ given by $j(x,s,t)=\ze(x,t)(s)$ that behaves +as follows on the boundary of the square for each fixed $x\in S^n$, where +$\tilde{g}_x(t)=\tilde{g}(x,t)$: +$$\diagram +\rrline^{c_{z_1}}|\tip & & \\ +& & \\ +\uuline^{c_{z_1}}|\tip \rrline_{h_x\cdot f_x^{-1}\cdot f_*\cdot h_*^{-1}}|\tip +& & \uuline_{e\com \tilde{g}_x}|\tip \\ +\enddiagram$$ +The desired homotopy $\tilde{h}$, written $\tilde{h}(x,s,t)$ where $s$ is the cone +coordinate and $t$ is the interval coordinate, should behave as follows on the boundary +of the square: +$$\diagram +\rrline^{e\com \tilde{g}_x}|\tip & & \\ +& & \\ +\uuline^{h_x}|\tip \rrline_{f_x}|\tip +& & \uuline_{h_*\cdot f_*^{-1}}|\tip \\ +\enddiagram$$ +Thus we can obtain $\tilde{h}$ by composing $j$ with a suitable +reparametrization $I^2\rtarr I^2$ of the square. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Show that, if $n\geq 2$, then $\pi_n(X\wed Y)$ is isomorphic to +$$\pi_n(X)\oplus\pi_n(Y)\oplus \pi_{n+1}(X\times Y, X\wed Y).$$ +\item Compute $\pi_n(\bR P^n,\bR P^{n-1})$ for $n\geq 2$. Deduce that the quotient map +$$(\bR P^n,\bR P^{n-1})\to (\bR P^n/\bR P^{n-1},*)$$ +does not induce an isomorphism of homotopy groups. +\item Compute the homotopy groups of complex projective space $\bC P^n$ in terms +of the homotopy groups of spheres. +\item Verify that the ``Hopf bundles'' are in fact bundles. +\item Show that $\pi_7(S^4)$ contains an element of infinite order. +\item Compute all of the homotopy groups of $\bR P^{\infty}$ and $\bC P^{\infty}$. +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{CW complexes} + +We introduce a large class of spaces, called CW complexes, between which a +weak equivalence is necessarily a homotopy equivalence. Thus, +for such spaces, the homotopy groups are, in a sense, a complete set of +invariants. Moreover, we shall see that every space is weakly +equivalent to a CW complex. + +\section{The definition and some examples of CW complexes} + +Let $D^{n+1}$ be the unit disk\index{disk} $\{ x\, |\, |x|\leq 1\} \subset \bR^{n+1}$ +with boundary $S^n$. + +\begin{defn} (i) A CW complex\index{CW complex} $X$ is a space $X$ which is the union of an +expanding sequence of subspaces $X^n$ such that, inductively, $X^0$ is +a discrete set of points (called vertices)\index{vertex} and $X^{n+1}$ is the pushout +obtained from $X^n$ by attaching disks $D^{n+1}$ along ``attaching maps''\index{attaching map} +$j: S^n\rtarr X^n$. Thus $X^{n+1}$ is the quotient space obtained from +$X^n\cup (J_{n+1}\times D^{n+1})$ by identifying $(j,x)$ with $j(x)$ for +$x\in S^n$, where $J_{n+1}$ is the discrete set of such attaching maps $j$. +Each resulting map +$D^{n+1}\rtarr X$ is called a ``cell.''\index{cell} The subspace $X^n$ is called the +$n$-skeleton of $X$.\index{skeleton!of a CW complex} + +(ii) More generally, given any space $A$, we +define a relative +CW complex\index{CW complex!relative} $(X,A)$ in the same fashion, but with +$X^0$ replaced by the union of +$A$ and a (possibly empty) discrete set of points; we write $(X,A)^n$, or $X^n$ +when $A$ is clear from the context, for the relative $n$-skeleton, and we say +that $(X,A)$ has dimension $\leq n$ if $X=X^n$. + +(iii) A subcomplex\index{subcomplex!of a CW complex} $A$ of a CW +complex $X$ is a subspace and a CW complex such that the composite of each +cell $D^n\rtarr A$ of $A$ and the inclusion of $A$ in $X$ is a cell of $X$. +That is, $A$ is the union of some of the cells of $X$. The pair $(X,A)$ can +then be viewed as a relative CW complex. + +(iv) A map of pairs $f:(X,A)\rtarr (Y,B)$ +between relative CW complexes is said to be ``cellular'' if $f(X^n)\subset Y^n$ +for all $n$.\index{cellular map} +\end{defn} + +Of course, pushouts and unions are understood in the topological sense, with the +compactly generated topologies. A subspace of $X$ is closed if and only if its +intersection with each $X^n$ is closed. + +\begin{exmps} (i) A graph is a one-dimensional CW complex. + +(ii) Via a homeomorphism $I\times I\iso D^2$, the standard presentations of the +torus\index{torus} +$T=S^1\times S^1$, the projective plane\index{projective plane} $\bR P^2$, and the +Klein bottle\index{Klein bottle} $K$ as quotients +of a square display these spaces as CW complexes with +one or two vertices, two edges, and one $2$-cell: + +\begin{small} +$$\diagram +& T & \\ +v \rrline^{e_1}|\tip \ddline_{e_2}|{\rotate\tip} & & v \ddline^{e_2}|{\rotate\tip} \\ +& & \\ +v \rrline_{e_1}|\tip & & v \\ +\enddiagram +\hspace{.2in} +\diagram +& \bR P^2 & \\ +v_1 \rrline^{e_1}|\tip \ddline_{e_2}|{\rotate\tip} & & v_2 \ddline^{e_2}|\tip \\ +& & \\ +v_2 \rrline_{e_1}|{\rotate\tip} & & v_1 \\ +\enddiagram +\hspace{.2in} +\diagram +& K & \\ +v \rrline^{e_1}|\tip \ddline_{e_2}|{\rotate\tip} & & v \ddline^{e_2}|\tip \\ +& & \\ +v \rrline_{e_1}|\tip & & v \\ +\enddiagram$$ +\end{small} + +(iii) For $n\geq 1$, $S^n$ is a CW complex with one vertex $\sset{*}$ and +one $n$-cell, the attaching map $S^{n-1}\rtarr \sset{*}$ being the only possible +map. Note that this entails a choice of homeomorphism $D^n/S^{n-1}\iso S^n$. If +$m\tip & & +X \uldashed^{\tilde{g}}|>\tip \llto^{i_1} \\ +\enddiagram $$ +\end{thm} +\begin{proof} +Proceed by induction over skeleta, applying the case $(D^n,S^{n-1})$ one +cell at a time to the $n$-cells of $X$ not in $A$. +\end{proof} + +In particular, if we take $e$ to be the identity map of $Y$, we see that the +inclusion $A\rtarr X$ is a cofibration. Observe that, by passage to colimits, +we are free to take $n=\infty$ in the theorem. + +We write $[X,Y]$ for homotopy classes of unbased maps in this chapter, and we +have the following direct and important application of HELP. + +\begin{thm}[Whitehead]\index{Whitehead theorem} If $X$ is a CW complex and $e:Y\rtarr Z$ is an +$n$-equivalence, then $e_*:[X,Y]\rtarr [X,Z]$ is +a bijection if \text{\em dim}\,$X 1$ and $X$ is not contractible, then it is known that $X$ has infinitely +many non-zero homotopy groups. The Whitehead theorem is thus surprisingly strong: in its +first statement, if low dimensional homotopy groups are mapped isomorphically, then so are +all higher homotopy groups. + +\section{The cellular approximation theorem} + +Cellular maps are under much better algebraic control than general maps, as will become +both clear and important later. Fortunately, any map between CW complexes is homotopic to +a cellular map. We need a lemma. + +\begin{defn} +A space $X$ is said to be $n$-connected\index{nconnected@$n$-connected space} if $\pi_q(X,x)=0$ +for $0\leq q\leq n$ and all $x$. +A pair $(X,A)$ is said to be $n$-connected if $\pi_0(A)\rtarr \pi_0(X)$ is surjective and +$\pi_q(X,A,a)=0$ for $1\leq q\leq n$ and all $a$. It is equivalent that the inclusion +$A\rtarr X$ be an $n$-equivalence. +\end{defn} + +\begin{lem} +A relative CW complex $(X,A)$ with no $m$-cells for $m\leq n$ is $n$-connected. In particular, +$(X,X^n)$ is $n$-connected for any CW complex $X$. +\end{lem} +\begin{proof} +Consider $f:(I^q,\pa I^q,J^q)\rtarr (X,A,a)$, where $q\leq n$. Since the image of $f$ +is compact, we may assume that $(X,A)$ has finitely many cells. By induction on the number +of cells, we may assume that $X=A\cup_jD^r$, where $r>n$. By smooth (or simplicial) +approximation, there is a map $f': I^q\rtarr X$ such that $f'=f$ on $\pa I^q$, +$f'\htp f \ \text{rel}\ \pa I^q$ and $f'$ misses a point $p$ in the interior of $D^r$. Clearly +we can deform $X-\sset{p}$ onto $A$ and so deform $f'$ to a map into $A$. +\end{proof} + +\begin{thm}[Cellular approximation]\index{cellular approximation theorem} +Any map $f:(X,A)\rtarr (Y,B)$ between relative CW complexes is homotopic relative to $A$ to +a cellular map. +\end{thm} +\begin{proof} +We proceed by induction over skeleta. To start the induction, note that any point of $Y$ +is connected by a path to a point in $Y^0$ and apply this to the images of points of +$X^0-A$ to obtain a homotopy of $f|X^0$ to a map into $Y^0$. Assume given +$g_n: X^n\rtarr Y^n$ and $h_n: X^n\times I\rtarr Y$ such +that $h_n: f|X^n\htp \io_n\com g_n$, where $\io_n: Y^n\rtarr Y$ +is the inclusion. For an attaching map $j: S^n\rtarr X^n$ of a cell +$\tilde{j}: D^{n+1}\rtarr X$, we apply HELP to the following diagram: +$$\diagram +S^n \ddto \rrto^{i_0} & & S^n\times I \dlto_{h_n\com (j\times\id)} \ddto & & S^n \llto_{i_1} +\ddto \dlto_{g_n\com j}\\ +& Y & & Y^{n+1} \llto_<(0.25){\io_{n+1}}\\ +D^{n+1} \urto^{f\com\tilde{j}} \rrto_{i_0} & & D^{n+1}\times I \uldashed_{h_{n+1}}|>\tip +& & D^{n+1} \llto^{i_1} \uldashed_{g_{n+1}}|>\tip \\ +\enddiagram$$ +where $g_n\com j: S^n\rtarr Y^n$ is composed with the inclusion $Y^n\rtarr Y^{n+1}$; +HELP applies since $\io_{n+1}$ is an $(n+1)$-equivalence. +\end{proof} + +\begin{cor} +For CW complexes $X$ and $Y$, any map $X \rtarr Y$ is homotopic to a +cellular map, and any two homotopic cellular maps are cellularly homotopic. +\end{cor} + +\section{Approximation of spaces by CW complexes} + +The following result says that there is a functor $\GA: h\sU \rtarr h\sU$ and +a natural transformation $\ga: \GA\rtarr \Id$ that assign a CW complex $\GA X$ +and a weak equivalence $\ga: \GA X\rtarr X$ to a space $X$. + +\begin{thm}[Approximation by CW complexes] For any space $X$, there +is a CW complex\, $\GA X$ and a weak equivalence $\ga:\GA X \rtarr X$. +For a map $f:X\rtarr Y$ and another such CW approximation $\ga: \GA Y\rtarr Y$, +there is a map $\GA f: \GA X\rtarr \GA Y$, unique up to homotopy, such that +the following diagram is homotopy commutative: +$$\diagram +\GA X \rto^{\GA f} \dto_{\ga} & \GA Y \dto^{\ga} \\ +X \rto_{f} & Y.\\ +\enddiagram$$ +If $X$ is $n$-connected, $n\geq 1$, then $\GA X$ can be chosen to have a unique vertex and no +$q$-cells for $1\leq q\leq n$. +\end{thm} +\begin{proof} The existence and uniqueness up to homotopy of $\GA f$ will be immediate +since the Whitehead theorem will give a bijection +$$\ga_*: [\GA X,\GA Y]\rtarr [\GA X,Y].$$ +Proceeding one path component at a time, we may as well assume that $X$ is path +connected, and we may then work with based spaces and based maps. We construct $\GA X$ as the +colimit of a sequence of cellular inclusions +$$\diagram +X_1\rto^{i_1} \ddrrto_{\ga_1} & X_2\rto^{i_2} \ddrto^{\ga_2} +& \cdots \rto & X_n\rto^{i_n} \ddlto_{\ga_n} \rto & X_{n+1} \ddllto^{\ga_{n+1}} \rto & \cdots \\ +& & & & & \\ +& & X. & & & \\ +\enddiagram$$ +Let $X_1$ be a wedge of spheres $S^q$, $q\geq 1$, one for each pair $(q,j)$, where +$j: S^q\rtarr X$ represents a generator of the group $\pi_q(X)$. On the $(q,j)$th +wedge summand, the map $\ga_1$ is the given map $j$. Clearly $\ga_1: X_1\rtarr X$ +induces an epimorphism on all homotopy groups. We give $X_1$ the CW structure induced +by the standard CW structures on the spheres $S^q$. Inductively, suppose that we have +constructed CW complexes $X_m$, cellular inclusions $i_{m-1}$, and maps $\ga_m$ for +$m\leq n$ such that $\ga_m\com i_{m-1}=\ga_{m-1}$ and $(\ga_m)_*:\pi_q(X_m)\rtarr \pi_q(X)$ +is a surjection for all $q$ and a bijection for $qn$ to arrange the +surjectivity of $\pi_*(X_0)\rtarr \pi_*(X)$. To construct $\GA f$, we first construct +it on $\GA A$ and then use HELP to extend to $\GA X$: +$$\diagram +\GA A \xto[0,3] \xto[3,0] \drto^{\ga} & & & \GA A\times I +\xto[3,0] \dlto_h & & & \GA A \xto[0,-3] \xto[3,0] \dlto_{\GA f}\\ +& A\rto^f \dto & B \dto & & & \GA B \dto \xto[0,-3]_{\ga} \dto & \\ +& X\rto_f & Y & & & \GA Y \xto[0,-3]^{\ga} & \\ +\GA X \urto^{\ga} \xto[0,3] & & & \GA X\times I +\uldashed_{\tilde{h}}|>\tip & & & \GA X \xto[0,-3] \uldashed_{\GA f}|>\tip \\ +\enddiagram$$ +The uniqueness up to homotopy of $\GA f$ is proved similarly. +\end{proof} + +\section{Approximation of excisive triads by CW triads} + +We will need another, and considerably more subtle, relative approximation theorem. +A triad $(X;A,B)$\index{triad} is a space $X$ together with subspaces $A$ and $B$. This must +not be confused with a triple $(X,A,B)$,\index{triple} which would require $B\subset A\subset X$. +A triad $(X;A,B)$ is said to be excisive\index{triad!excisive} if $X$ is the union of the +interiors of $A$ and $B$. Such triads play a fundamental role in homology and cohomology theory, +and some version of the arguments to follow must play a role in any treatment. +We prefer to use these arguments to prove a strong homotopical result, rather than +its pale homological reflection that is seen in standard treatments of the subject. + +A CW triad $(X;A,B)$\index{triad!CW} is a CW complex $X$ with subcomplexes $A$ and $B$ such that +$X=A\cup B$. + +\begin{thm} +Let $(X;A,B)$ be an excisive triad and let $C=A\cap B$. Then there is a CW triad +$(\GA X;\GA A,\GA B)$ and a map of triads +$$\ga: (\GA X;\GA A,\GA B)\rtarr (X;A,B)$$ +such that, with $\GA C = \GA A\cap \GA B$, the maps +$$ \ga: \GA C\rtarr C,\ \ \ga: \GA A\rtarr A,\ \ \ga: \GA B\rtarr B,\ \ \tand \ \ +\ga: \GA X\rtarr X$$ +are all weak equivalences. If $(A,C)$ is $n$-connected, then $(\GA A,\GA C)$ can be +chosen to have no $q$-cells for $q\leq n$, and similarly for $(B,C)$. Up to homotopy, +CW approximation of excisive triads is functorial in such a way that $\ga$ is natural. +\end{thm} +\begin{proof} +Choose a CW approximation $\ga: \GA C\rtarr C$ and use the previous +result to extend it to CW approximations +$$\ga: (\GA A,\GA C)\rtarr (A,C) \ \ \ \tand \ \ \ \ga: (\GA B,\GA C)\rtarr (B,C).$$ +We then define $\GA X$ to be the pushout $\GA A\cup_{\GA C}\GA B$ and let $\ga: \GA X\rtarr X$ +be given by the universal property of pushouts. Certainly $\GA C=\GA A\cap \GA B$. All of the +conclusions except for the assertion that $\ga: \GA X\rtarr X$ is a weak equivalence follow +immediately from the result for pairs, and the lemma and theorem below will complete the proof. +\end{proof} + +A CW triad $(X;A,B)$ is not excisive, since $A$ and $B$ are closed in $X$, but it is equivalent +to an excisive triad. To see this, +we describe a simple but important general construction. Suppose that maps $i: C\rtarr A$ +and $j: C\rtarr B$ are given. Define the double mapping cylinder +$$M(i,j)= A\cup (C\times I)\cup B$$ +to be the space obtained from $C\times I$ by gluing $A$ to $C\times\sset{0}$ along $i$ and +gluing $B$ to $C\times\sset{1}$ along $j$. Let $A\cup_C B$ denote the pushout of $i$ and $j$ +and observe that we obtain a natural quotient map $q: M(i,j)\rtarr A\cup_C B$ by collapsing +the cylinder, sending $(c,t)$ to the image of $c$ in the pushout. + +\begin{lem} +For a cofibration $i: C\rtarr A$ and any map $j: C\rtarr B$, the quotient map +$q: M(i,j)\rtarr A\cup_C B$ is a homotopy equivalence. +\end{lem} +\begin{proof} +Because $i$ is a cofibration, the retraction $r: Mi\rtarr A$ is a cofiber homotopy +equivalence. That is, there is a homotopy inverse map and a pair of homotopies under $C$. +These maps and homotopies induce maps of the pushouts that are obtained by gluing $B$ to +$Mi$ and to $C$, and $q$ is induced by $r$. +\end{proof} + +When $i$ is a cofibration and $j$ is an inclusion, with $X=A\cup B$ and $C=A\cap B$, +we can think of $q$ as giving a map of triads +$$ q: (M(i,j); A\cup(C\times[0,2/3)), (C\times (1/3,1])\cup B)\rtarr (A\cup_C B;A,B).$$ +The domain triad is excisive, and $q$ restricts to homotopy equivalences from the domain +subspaces and their intersection to the target subspaces $A$, $B$, and $C$. This applies +when $(X;A,B)$ is a CW triad with $C=A\cap B$. Now our theorem on the approximation of +excisive triads is a consequence of the following result. + +\begin{thm} +If $e: (X;A,B)\rtarr (X';A',B')$ is a map of excisive triads such that the maps +$$e: C\rtarr C',\ \ \ e: A\rtarr A',\ \ \ \tand \ \ \ e: B\rtarr B'$$ +are weak equivalences, where $C=A\cap B$ and $C'=A'\cap B'$, then $e: X\rtarr X'$ +is a weak equivalence. +\end{thm} +\begin{proof} +By our technical lemma giving equivalent conditions for a map $e$ to be a weak +equivalence, it suffices to show that if $f|S^n=e\com g$ in the following diagram, +then there exists a map $\tilde{g}$ such that $\tilde{g}|S^n = g$ and +$f\htp e\com \tilde{g}\ \text{rel} \ S^n$: +$$\diagram +X\rto^e & X' \\ +S^n \uto^g \rto & D^{n+1} \uldashed_{\tilde{g}}|>\tip \uto_f. \\ +\enddiagram$$ +We may assume without loss of generality that $S^n\subset U\subset D^{n+1}$, where $U$ is +open in $D^{n+1}$ and $g$ is the restriction of a map $\hat{g}: U\rtarr X$ such that +$f|U=e\com \hat{g}$. To see this, define a deformation $d: D^{n+1}\times I\rtarr D^{n+1}$ by +$$d(x,t)=\begin{cases} +2x/(2-t) \ \ \ \text{if}\ \ |x|\leq (2-t)/2\\ +x/|x| \ \ \ \ \ \ \ \ \ \ \text{if}\ \ |x|\geq (2-t)/2. +\end{cases}$$ +Then $d(x,0)=x$, $d(x,t)=x$ if $x\in S^n$, and $d_1$ maps the boundary collar +$\sset{x\ |\ |x|\geq 1/2}$ onto $S^n$. Let $U$ be the open boundary collar +$\sset{x\ |\ |x|> 1/2}$. Define $\hat{g}=g\com d_1: U\rtarr X$ and define +$f'=f\com d_1: D^{n+1}\rtarr X'$. Then $\hat{g}|S^n = g$, $e\com \hat{g}=f'|U$, +and $f'\htp f \ \text{rel}\ S^n$. Thus the conclusion will hold for $f$ if it holds +with $f$ replaced by $f'$. + +With this assumption on $g$ and $f$, we claim first that the closed sets +$$C_A=g^{-1}(X-\text{int}\,A)\cup \overline{f^{-1}(X'-A')}$$ +and +$$C_B=g^{-1}(X-\text{int}\,B)\cup \overline{f^{-1}(X'-B')},$$ +have empty intersection. Indeed, these sets are contained in the sets +$\hat{C}_A$ and $\hat{C}_B$ that are obtained by replacing $g$ by $\hat{g}$ in the +definitions of $C_A$ and $C_B$, and we claim that $\hat{C}_A\cap\hat{C}_B=\emptyset$. Certainly +$$\hat{g}^{-1}(X-\text{int}\,A)\cap \hat{g}^{-1}(X-\text{int}\,B)=\emptyset$$ +since $(X-\text{int}\,A)\cap (X-\text{int}\,B)=\emptyset$. Similarly, +$$f^{-1}(X'-\text{int}\,A')\cap f^{-1}(X'-\text{int}\,B')=\emptyset.$$ +Since $\overline{f^{-1}(X'-A')}\subset f^{-1}(X'-\text{int}\,A')$ and similarly +for $B$, this implies that +$$\overline{f^{-1}(X'-A')}\cap \overline{f^{-1}(X'-B')}=\emptyset.$$ +Now suppose that $v\in \hat{C}_A\cap\hat{C}_B$. In view of the possibilities that we have ruled +out, we may assume that +$$v\in \hat{g}^{-1}(X-\text{int}\,A)\cap \overline{f^{-1}(X'-B')}\subset +\hat{g}^{-1}(\text{int}\,B)\cap \overline{f^{-1}(X'-B')}.$$ +Since $\hat{g}^{-1}(\text{int}\,B)$ is an open subset of $D^n$, there must be a point +$$u\in \hat{g}^{-1}(\text{int}\,B) \cap f^{-1}(X'-B').$$ +Then $\hat{g}(u)\in \text{int}\,B\subset B$ but $f(u)\not\in B'$. This contradicts +$f|U=e\com\hat{g}$. + +We can subdivide $D^{n+1}$ sufficiently finely (as a simplicial or CW complex) that no cell +intersects both $C_A$ and $C_B$. Let $K_A$ be the union of those cells $\si$ such that +$$g(\si\cap S^n)\subset\ \text{int}\,A\ \ \tand \ \ f(\si)\subset\ \text{int}\,A'$$ +and define $K_B$ similarly. If $\si$ does not intersect $C_A$, then $\si\subset K_A$, and if +$\si$ does not intersect $C_B$, then $\si\subset K_B$. Therefore $D^{n+1}=K_A\cup K_B$. By +HELP, we can obtain a map $\bar{g}$ such that the lower triangle in the diagram +$$\diagram +A\cap B \rto^e & A'\cap B' \\ +S^n\cap (K_A\cap K_B) \uto^g \rto & K_A\cap K_B \uto_f \uldashed_{\bar{g}}|>\tip \\ +\enddiagram$$ +commutes, together with a homotopy $\bar{h}: (K_A\cap K_B)\times I\rtarr A'\cap B'$ +such that +$$\bar{h}: f\htp e\com\bar{g}\ \text{rel}\,S^n\cap (K_A \cap K_B).$$ +Define $\bar{g}_A: K_A\cap(S^n\cup K_B)\rtarr A$ to be $g$ on $K_A\cap S^n$ and $\bar{g}$ on +$K_A\cap K_B$. Since $f=e\com g$ on $K_A\cap S^n$ and $\bar{h}: f\htp e\com\bar{g}$ on +$K_A\cap K_B$, $\bar{h}$ induces a homotopy +$$\bar{h}_A: f|K_A\cap(S^n\cup B)\htp e\com g_A\ \text{rel}\,S^n\cap K_A .$$ +Applying HELP again, we can obtain maps $\tilde{g}_A$ and $\tilde{h}_A$ such that the +following diagram commutes: +\begin{small} +$$ +\diagram +K_A\cap(S^n\cup K_B) \ddto \rrto^{i_0} & & K_A\cap(S^n\cup K_B) \times I \dlto_{\bar{h}_A} \ddto +& & K_A\cap(S^n\cup K_B) \llto_{i_1} \dlto_{\bar{g}_A} \ddto \\ +& A' & & A \llto_<(0.4){e} & \\ +K_A \rrto_{i_0} \urto^{f} && K_A \times I \uldashed^{\tilde{h}_A}|>\tip & & +K_A \uldashed^{\tilde{g}_A}|>\tip \llto^{i_1} \\ +\enddiagram $$ +\end{small} +We have a symmetric diagram with the roles of $K_A$ and $K_B$ reversed. The maps $\tilde{g}_A$ +and $\tilde{g}_B$ agree on $K_A\cap K_B$ and together define the desired map +$\tilde{g}: D^{n+1}\rtarr X$. The homotopies $\tilde{h}_A$ and $\tilde{h}_B$ agree on +$(K_A\cap K_B)\times I$ and together define the desired homotopy +$\tilde{h}_A: f\htp e\com \tilde{g}\ \text{rel}\,S^n$. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Show that complex projective space $\bC P^n$ is a CW complex with one $2q$-cell for each $q$, +$0\leq q\leq n$. +\item Let $X = \sset{x| x = 0 \ \text{or}\ x = 1/n \ \text{for a positive integer $n$}}\subset \bR$. +Show that $X$ does not have the homotopy type of a CW complex. +\item Assume given maps $f: X\rtarr Y$ and $g: Y\rtarr X$ such that $g\com f$ is homotopic to +the identity. (We say that $Y$ ``dominates'' $X$.) Suppose that $Y$ is a CW complex. Prove +that $X$ has the homotopy type of a CW complex. +\end{enumerate} + +Define the Euler characteristic\index{Euler characteristic!of a CW complex} $\ch (X)$ of a +finite CW complex $X$ to be the alternating +sum $\sum(-1)^n\ga_n(X)$, +where $\ga_n(X)$ is the number of $n$-cells of $X$. Let $A$ be a subcomplex of a CW complex $X$, +let $Y$ be a CW complex, let $f: A\rtarr Y$ be a cellular map, and let $Y\cup_f X$ be the +pushout of $f$ and the inclusion $A\rtarr X$. + +\begin{enumerate} +\item[4.] Show that $Y\cup_f X$ is a CW complex with $Y$ as a subcomplex and +$X/A$ as a quotient complex. Formulate and prove a formula relating the Euler characteristics +$\ch(A)$, $\ch(X)$, $\ch(Y)$, and $\ch(Y\cup_fX)$ when $X$ and $Y$ are finite. +\item[5.]* Think about proving from what we have done so far that $\ch(X)$ depends only on the +homotopy type of $X$, not on its decomposition as a finite CW complex. +\end{enumerate} + +\chapter{The homotopy excision and suspension theorems} + +The fundamental obstruction to the calculation of homotopy groups is +the failure of excision: for an excisive triad $(X;A,B)$, the inclusion +$(A,A\cap B)\rtarr (X,B)$ fails to induce an isomorphism of homotopy +groups in general. It is this that distinguishes homotopy groups from +the far more computable homology groups. However, we do have such an +isomorphism in a range of dimensions. This implies the Freudenthal +suspension theorem, which gives that $\pi_{n+q}(\SI^n X)$ is independent +of $n$ if $q$ is small relative to $n$. We shall rely on the consequence +$\pi_n(S^n)\iso\bZ$ in our construction of homology groups. + +\section{Statement of the homotopy excision theorem} + +We shall prove the following theorem later in this chapter, but we first +explain its consequences. + +\begin{defn} +A map $f:(A,C)\rtarr (X,B)$ +of pairs is an $n$-equivalence,\index{nequivalence@$n$-equivalence} $n\geq 1$, if +$$(f_*)^{-1}(\im(\pi_0(B)\rtarr \pi_0(X)))=\im(\pi_0(C)\rtarr \pi_0(A))$$ +(which holds automatically when $A$ and $X$ are path connected) and, for all +choices of basepoint in $C$, +$$f_*:\pi_q(A,C )\rtarr \pi_q(X,B)$$ +is a bijection for $qn. +\end{cases}$$ + +This calculation well illustrates general facts about the homology of compact connected +closed $n$-manifolds $M$ that we shall prove later. The $n$th integral homology group of +such a manifold $M$ is $\bZ$ if $M$ is orientable and zero if $M$ is not orientable. The +$n$th mod $2$ homology group of $M$ is $\bZ_2$ whether or not $M$ is orientable. + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item If $X$ is a finite CW complex, show that $\ch(X)=\ch(H_*(X;k))$ for any field $k$. +\item Let $A$ be a subcomplex of a CW complex $X$, let $Y$ be a CW complex, and let +$f: A\rtarr Y$ be a cellular map. What is the relationship between $H_*(X,A)$ and +$H_*(Y\cup_fX,Y)$? Is there a similar relationship between $\pi_*(X,A)$ and +$\pi_*(Y\cup_fX,Y)$? If not, give a counterexample. +\item Fill in the details of the computation of the differentials on the cellular +chains in the examples in \S5. +\item Compute $H_*(S^m\times S^n)$ for $m\geq 1$ and $n\geq 1$. Convince yourself that +you can do this by use of CW structures, by direct deduction from the axioms, and by +the K\"{u}nneth theorem (for which see Chapter 17). +\item Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group +of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\bC^n$, +$n \geq 2$. Then $\pi\subset S^1$ acts freely on $S^{2n-1}$ via +$$\ze (z_1,\ldots\!, z_n) = (\ze z_1,\ldots\!,\ze z_n). $$ +Let $L^n = S^{2n-1}/\pi$ be the orbit space; it is called a lens space and is an odd primary +analogue of $\bR P^n$. The obvious quotient map $S^{2n-1}\rtarr L^n$ is a universal covering. +\begin{enumerate} +\item[(a)] Compute the integral homology of $L^n$, $n\geq 2$, by mimicking the calculation of +$H_*(\bR P^n)$. +\item[(b)] Compute $H_*(L^n;\bZ_p)$, where $\bZ_p = \bZ/p\bZ$. +\end{enumerate} +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{Derivations of properties from the axioms} + +Returning to the axiomatic approach to homology, we assume given a theory on pairs +of spaces and make some deductions from the axioms. We abbreviate notations by setting +$E_q(X,A)=H_q(X,A;\pi)$. However, the arguments in this chapter make no use whatever +of the dimension axiom. A ``generalized homology theory''\index{homology theory!generalized} $E_*$ +is defined to be a +system of functors $E_q(X,A)$ and natural transformations $\pa:E_q(X,A)\rtarr E_{q-1}(A)$ +that satisfy all of our axioms except for the dimension axiom. Similarly, we have the +notion of a generalized homology theory on CW pairs, and the results of the first section +of the previous chapter generalize directly to give the following result. + +\begin{thm} A homology theory $E_*$ on pairs of spaces determines and is determined +by its restriction to a homology theory $E_*$ on pairs of CW complexes. +\end{thm} + +The study of such generalized homology theories pervades modern algebraic topology, +and we shall describe some examples later on. The brave reader may be willing to think +of $E_*$ in such generality in this chapter. The timorous reader may well prefer to +think of $E_*(X,A)$ concretely, following our proposal that $E_*(X,A)$ be taken as +an alternative notation for $H_*(X,A;\pi)$. + +\section{Reduced homology; based versus unbased spaces} + +One of the themes of this chapter is the relationship between homology theories on +pairs of spaces and reduced homology theories on based spaces. The latter are more +convenient in most advanced work in algebraic topology. For a based space $X$, we +define the reduced homology\index{reduced homology} of $X$ to be +$$\tilde{E}_q(X)=E_q(X,*).$$ +Since the basepoint is a retract of $X$, there results a direct sum +decomposition +$$ E_*(X) \iso \tilde{E}_*(X)\oplus E_*(*)$$ +that is natural with respect to based maps. For $*\in A\subset X$, the summand $E_*(*)$ +maps isomorphically under the map $E_*(A)\rtarr E_*(X)$, and the exactness axiom implies +that there is a reduced long exact sequence +$$\cdots\rtarr \tilde{E}_q(A)\rtarr \tilde{E}_q(X)\rtarr +E_q(X,A)\overto{\pa} \tilde{E}_{q-1}(A)\rtarr \cdots .$$ + +We can obtain the unreduced homology groups as special cases of the reduced ones. For an +unbased space $X$, we define a based space $X_+$ by adjoining a disjoint basepoint +to $X$. By the additivity axiom, we see immediately that +$$ E_*(X)=\tilde{E}_*(X_+).$$ +Similarly, a map $f: X\rtarr Y$ of unbased spaces induces a map $f_+: X_+\rtarr Y_+$ +of based spaces, and the map $f_*$ on unreduced homology coincides with the map +$(f_+)_*$ on reduced homology. + +We shall make considerable use of cofiber sequences in this chapter. +To be consistent about this, we should always work with reduced cones and +cofibers. However, it is more convenient to make the convention that we work with +unreduced cones and cofibers when we apply unreduced homology theories, and we work with +reduced cones and cofibers when we apply reduced homology theories. In fact, the +unreduced cone\index{cone!unreduced} on a space $Y$ coincides with the reduced +cone\index{cone!reduced} on $Y_+$: the line through +the disjoint basepoint is identified to the cone point when constructing the reduced cone +on $Y_+$. Therefore the unreduced cofiber of an unbased map $f$ coincides with the reduced +cofiber of the based map $f_+$. Our convention really means that we are always working with +reduced cofibers, but when we are studying unreduced homology theories we are implicitly +applying the functor $(-)_+$ to put ourselves in the based context before constructing +cones and cofibers. + +The observant reader will have noticed that the unreduced suspension\index{suspension!unreduced} of +$X$ is {\em not} the reduced suspension\index{suspension!reduced} on $X_+$. Rather, under either +interpretation of suspension, $\SI(X_+)$ is homotopy equivalent to the wedge of $\SI(X)$ and a +circle. + +\section{Cofibrations and the homology of pairs} + +We use cofibrations to show that the homology of pairs of spaces is in principle a special +case of the reduced homology of spaces. + +\begin{thm} +For any cofibration\index{cofibration} $i: A\rtarr X$, the quotient map $q: (X,A)\rtarr (X/A,*)$ +induces an isomorphism +$$E_*(X,A)\rtarr E_*(X/A,*)=\tilde{E}_*(X/A).$$ +\end{thm} +\begin{proof} +Consider the (unreduced) cofiber +$$Ci= X\cup_iCA = X\cup_i A\times I/A\times\sset{1}.$$ +We have an excisive triad +$$(Ci; X\cup_i A\times [0,2/3], A\times [1/3,1]/A\times \sset{1}).$$ +The excision axiom gives that the top inclusion in the following +commutative diagram induces an isomorphism on passage to homology: +$$\diagram +(X\cup_i A\times [0,2/3], A\times [1/3,2/3])\rto \dto_r +& (Ci,A\times [1/3,1]/A\times \sset{1}) \dto^{\psi}\\ +(X,A)\rto_q & (X/A,*)\\ +\enddiagram$$ +The map $r$ is obtained by restriction of the retraction $Mi\rtarr X$ and +is a homotopy equivalence of pairs. The map $\psi$ collapses $CA$ to a point +and is also a homotopy equivalence of pairs. The conclusion follows. +\end{proof} + +As in our construction of cellular homology, we choose a homotopy inverse +$\ps^{-1}: X/A\rtarr Ci$ and consider the composite +$$X/A\overto{\ps^{-1}} Ci\overto{\pi} \SI A$$ +to be a topological boundary map +$$\pa: X/A\rtarr \SI A.$$ +Observe that we may replace any inclusion $i: A\rtarr X$ by the canonical +cofibration $A\rtarr Mi$ and then apply the result just given to obtain +$$E_*(X,A)\iso \tilde{E}_*(Ci).$$ + +\section{Suspension and the long exact sequence of pairs} + +We have a fundamentally important consequence of the results of the previous +section, which should be contrasted with what happened with homotopy groups. +Recall that a basepoint $*\in X$ is nondegenerate if the inclusion +$\sset{*}\rtarr X$ is a cofibration. This ensures that the inclusion of +the line through the basepoint in the unreduced suspension of $X$ is a +cofibration, so that the map from the unreduced suspension to the suspension +that collapses out the line through the basepoint is a homotopy equivalence. +We apply reduced homology here, so we use reduced cones and suspensions. + +\begin{thm} For a nondegenerately based space $X$, there is a natural isomorphism +$$\SI: \tilde{E}_q(X)\iso \tilde{E}_{q+1}(\SI X).$$ +\end{thm} +\begin{proof} Since $CX$ is contractible, its reduced homology is identically +zero. By the reduced long exact sequence, there results an isomorphism +$$\diagram \tilde{E}_{q+1}(\SI X)\iso \tilde{E}_{q+1}(CX/X) +\overto{\pa} \tilde{E}_q(X). \qed +\enddiagram$$ +\renewcommand{\qed}{}\end{proof} + +An easy diagram chase gives the following consequence, which describes the axiomatically +given connecting homomorphism of the pair $(X,A)$ in terms of the topological boundary +map\index{topological boundary map} +$\pa: X/A\rtarr \SI A$ and the suspension isomorphism. + +\begin{cor} Let $*\in A\subset X$, where $i: A\rtarr X$ is a cofibration between +nondegenerately based spaces. In the long exact sequence +$$\cdots\rtarr \tilde{E}_q(A)\rtarr \tilde{E}_q(X)\rtarr +\tilde{E}_q(X/A)\overto{\pa} \tilde{E}_{q-1}(A)\rtarr \cdots $$ +of the pair $(X,A)$, the connecting homomorphism $\pa$ is the composite +$$\tilde{E}_q(X/A)\overto{\pa_*} \tilde{E}_{q}(\SI A)\overto{\SI^{-1}} \tilde{E}_{q-1}(A).$$ +\end{cor} + +Since $S^0$ consists of two points, $\tilde{E}_*(S^0)=E_*(*)$. Since $S^n$ is the +suspension of $S^{n-1}$, we have the following special case of the suspension +isomorphism. + +\begin{cor} +For any $n$ and $q$, +$$\tilde{E}_q(S^n)\iso E_{q-n}(*).$$ +\end{cor} + +Of course, for the theory $H_*(X;\pi)$, this was immediate from our construction in +terms of cellular chains. + +\section{Axioms for reduced homology} + +In the study of generalized homology theories, it is most convenient to +restrict attention to reduced homology theories\index{homology theory!reduced} defined +on nondegenerately +based spaces. The results of the previous sections imply that we can do so +without loss of generality. Again the reader has the choice of bravery or +timorousness in interpreting $E_*$, but we opt for bravery: + +\begin{defn} A reduced homology theory $\tilde{E}_*$ consists of functors +$\tilde{E}_q$ from the homotopy category of nondegenerately based spaces +to the category of Abelian groups that satisfy the following axioms. +\begin{itemize} +\item EXACTNESS\index{exactness axiom}\ \ If $i: A\rtarr X$ is a cofibration, then the sequence +$$\tilde{E}_q(A)\rtarr \tilde{E}_q(X)\rtarr +\tilde{E}_q(X/A)$$ +is exact. +\item SUSPENSION\index{suspension axiom}\ \ +For each integer $q$, there is a natural isomorphism +$$\SI: \tilde{E}_q(X)\iso \tilde{E}_{q+1}(\SI X).$$ +\item ADDITIVITY\index{additivity axiom}\ \ +If $X$ is the wedge of a set of nondegenerately based spaces $X_i$, then +the inclusions $X_i\rtarr X$ induce an isomorphism +$$\textstyle{\sum}_i \tilde{E}_*(X_i)\rtarr \tilde{E}_*(X).$$ +\item WEAK EQUIVALENCE\index{weak equivalence axiom}\ \ If $f:X\rtarr Y$ is a weak equivalence, then +$$f_*: \tilde{E}_*(X)\rtarr \tilde{E}_*(Y)$$ +is an isomorphism. +\end{itemize} +\end{defn} + +The reduced form of the dimension axiom would read +$$\tilde{H}_0(S^0)=\pi \ \ \tand \ \ \tilde{H}_q(S^0)=0\ \text{for}\ q\neq 0.$$ + +\begin{thm} A homology theory $E_*$ on pairs of spaces determines and is +determined by a reduced homology theory $\tilde{E}_*$ on nondegenerately +based spaces. +\end{thm} +\begin{proof} +Given a theory on pairs, we define $\tilde{E}_*(X)=E_*(X,*)$ and deduce +the new axioms. For additivity, the specified wedge is the quotient +$(\amalg X_i)/(\amalg\sset{*_i})$, where $*_i$ is the basepoint of $X_i$, and our +result on quotients of cofibrations applies to compute its homology. Conversely, +assume given a reduced homology theory $\tilde{E}_*$, and define +$$E_*(X)=\tilde{E}_*(X_+) \ \ \tand \ \ E_*(X,A)=\tilde{E}_*(C(i_+)),$$ +where $C(i_+)$ is the cofiber of the based inclusion $i_+: A_+\rtarr X_+$. +Equivalently, $C(i_+)$ is the unreduced cofiber of $i: A\rtarr X$ with its +cone point as basepoint. We must show that the suspension axiom and our restricted +exactness axiom imply the original, seemingly much stronger, exactness and excision +axioms. We have the long exact cofiber sequence associated to the based inclusion +$i_+: A_+\rtarr X_+$, in which each consecutive pair of maps is equivalent to a +cofibration and the associated quotient map. Noting that $X_+/A_+=X/A$, we define +the connecting homomorphism $\pa_q: E_q(X,A)\rtarr E_{q-1}(A)$ to be the composite +$$\tilde{E}_q(X_+/A_+)\overto{\pa_*}\tilde{E}_q(\SI A_+)\overto{\SI^{-1}}\tilde{E}_{q-1}(A_+)$$ +and find that the exactness and suspension axioms for $\tilde{E}_*$ imply the exactness +axiom for $E_*$. For excision, we could carry out a similarly direct homotopical argument, +but it is simpler to observe that this follows from the equivalence of theories on pairs of +spaces with theories on pairs of CW complexes together with the next two theorems. For the +additivity axiom, we note that the cofiber of a disjoint union of maps is the wedge of the +cofibers of the given maps. +\end{proof} + +\begin{cor} For nondegenerately based spaces $X$, $E_*(X)$ is naturally isomorphic to +$\tilde{E}_*(X)\oplus E_*(*)$. +\end{cor} +\begin{proof} +The long exact sequence in $E_*$ of the pair $(X,*)$ is naturally split in each degree +by means of the homomorphism induced by the projection $X\rtarr \sset{*}$. +\end{proof} + +We require of based CW complexes that the basepoint be a vertex. It is certainly +a nondegenerate basepoint. We give the circle its standard CW structure and so +deduce a CW structure on the suspension of a based CW complex. + +\begin{defn} A reduced homology theory\index{homology theory!reduced} $\tilde{E}_*$ on based +CW complexes consists +of functors $\tilde{E}_q$ from the homotopy category of based CW complexes +to the category of Abelian groups that satisfy the following axioms. +\begin{itemize} +\item EXACTNESS\index{exactness axiom}\ \ If $A$ is a subcomplex of $X$, then the sequence +$$\tilde{E}_q(A)\rtarr \tilde{E}_q(X)\rtarr +\tilde{E}_q(X/A)$$ +is exact. +\item SUSPENSION\index{suspension axiom} \ \ +For each integer $q$, there is a natural isomorphism +$$\SI: \tilde{E}_q(X)\iso \tilde{E}_{q+1}(\SI X).$$ +\item ADDITIVITY\index{additivity axiom}\ \ +If $X$ is the wedge of a set of based CW complexes $X_i$, then +the inclusions $X_i\rtarr X$ induce an isomorphism +$$\textstyle{\sum}_i \tilde{E}_*(X_i)\rtarr \tilde{E}_*(X).$$ +\end{itemize} +\end{defn} + +\begin{thm} A reduced homology theory $\tilde{E}_*$ on nondegenerately based spaces +determines and is determined by its restriction to a reduced homology theory on +based CW complexes. +\end{thm} +\begin{proof} +This is immediate by CW approximation of based spaces. +\end{proof} + +\begin{thm} +A homology theory $E_*$ on CW pairs determines and is determined by a reduced +homology theory $\tilde{E}_*$ on based CW complexes. +\end{thm} +\begin{proof} +Given a theory on pairs, we define $\tilde{E}_*(X)=E_*(X,*)$ and deduce +the new axioms directly. Conversely, given a reduced theory on based CW +complexes, we define +$$ E_*(X)=\tilde{E}_*(X_+)\ \ \tand\ \ E_*(X,A)=\tilde{E}_*(X/A).$$ +Of course $X/A$ is homotopy equivalent to $C(i_+)$, where $i_+: A_+\rtarr X_+$ is the +inclusion. The arguments for exactness and additivity are the same as those given in the +analogous result for nondegenerately based spaces, but now excision is obvious since if +$(X;A,B)$ is a CW triad, then the inclusion $A/A\cap B \rtarr X/B$ is an isomorphism of +based CW complexes. +\end{proof} + +\section{Mayer-Vietoris sequences} + +The Mayer-Vietoris sequences are long exact sequences associated to excisive triads +that will play a fundamental role in our later proof of the Poincar\'{e} duality theorem. +We need two preliminaries, both of independent interest. The first is the long exact +sequence of a triple $(X,A,B)$ of spaces $B\subset A\subset X$, which is just like its +analogue for homotopy groups. + +\begin{prop} For a triple $(X,A,B)$, the following sequence\index{triple!exact sequence of} is exact: +$$\cdots \rtarr E_q(A,B)\overto{i_*} E_q(X,B)\overto{j_*} E_q(X,A) \overto{\pa} +E_{q-1}(A,B)\rtarr \cdots.$$ +Here $i:(A,B)\rtarr (X,B)$ and $j:(X,B)\rtarr (X,A)$ are inclusions and $\pa$ is the +composite +$$E_q(X,A)\overto{\pa}E_{q-1}(A)\rtarr E_{q-1}(A,B).$$ +\end{prop} +\begin{proof} +There are two easy arguments. One can either use diagram chasing from the various long +exact sequences of pairs or one can apply CW approximation to replace $(X,A,B)$ by a +triple of CW complexes. After the replacement, we have that $X/A\iso(X/B)/(A/B)$ as a CW complex, +and the desired sequence is the reduced exact sequence of the pair $(X/B,A/B)$. +\end{proof} + +\begin{lem} +Let $(X;A,B)$ be an excisive triad and set $C=A\cap B$. The map +$$E_*(A,C)\oplus E_*(B,C)\rtarr E_*(X,C)$$ +induced by the inclusions of $(A,C)$ and $(B,C)$ in $(X,C)$ is an isomorphism. +\end{lem} +\begin{proof} +Again, there are two easy proofs. One can either pass to homology from the diagram +$$\diagram +(B,C) \ddto_{excision} \drto & & (A,C) \dlto \ddto^{excision} \\ +& (X,C) \drto \dlto & \\ +(X,A) & & (X,B) \\ +\enddiagram$$ +and use algebra or one can approximate $(X;A,B)$ by a CW triad, for which +$$X/C\iso A/C\wed B/C$$ +as a CW complex. +\end{proof} + +\begin{thm}[Mayer-Vietoris sequence]\index{Mayer-Vietoris sequence} Let +$(X;A,B)$ be an excisive triad and set $C=A\cap B$. +The following sequence is exact: +$$\cdots \rtarr E_q(C)\overto{\ps} E_q(A)\oplus E_q(B)\overto{\ph} E_q(X)\overto{\DE} +E_{q-1}(C)\rtarr \cdots.$$ +Here, if $i: C\rtarr A$, $j: C\rtarr B$, $k: A\rtarr X$, and $\ell: B\rtarr X$ +are the inclusions, then +$$\psi(c)=(i_*(c),j_*(c)), \ \ \ \ \ph(a,b)= k_*(a)-\ell_*(b),$$ +and $\DE$ is the composite +$$E_q(X)\rtarr E_q(X,B)\iso E_q(A,C)\overto{\pa} E_{q-1}(C).$$ +\end{thm} +\begin{proof} Note that the definition of $\ph$ requires a sign in order to make $\ph\com\ps = 0$. +The proof of exactness is algebraic diagram chasing and is left as an exercise. The following +diagram may help: +$$\diagram +& & E_q(C) \dlto \ddto^{i_*} \drto & & \\ +& E_q(B) \drto & & E_q(A) \dlto & \\ +& & E_q(X) \dlto \ddto^{j_*} \drto \xto '[0,2] '[4,2]^{\DE} '[4,1] +\xto '[0,-2] '[4,-2]_{-\DE} '[4,-1] & & \\ +& E_q(X,A) & & E_q(X,B) & \\ +& & E_q(X,C) \ulto \ddto^{\pa} \urto & & \\ +& E_q(B,C) \drto_{\pa} \urto \uuto^{\iso} & & E_q(A,C) \dlto^{\pa} \ulto \uuto_{\iso} & \\ +& & E_{q-1}(C) & & \\ +\enddiagram$$ +Here the arrow labeled ``$-\DE$'' is in fact $-\DE$ by an algebraic argument from the direct +sum decomposition of $E_q(X,C)$. Alternatively, one can use CW approximation. For a CW triad, +there is a short exact sequence +$$0\rtarr C_*(C)\rtarr C_*(A)\oplus C_*(B)\rtarr C_*(X)\rtarr 0$$ +whose associated long exact sequence is the Mayer-Vietoris sequence. +\end{proof} + +We shall also need a relative analogue, but the reader may wish to ignore this for now. It +will become important when we study manifolds with boundary. + +\begin{thm}[Relative Mayer-Vietoris sequence]\index{Mayer-Vietoris sequence!relative} Let +$(X;A,B)$ be an excisive \linebreak +triad and set +$C=A\cap B$. Assume that $X$ is contained in some ambient space $Y$. The following +sequence is exact: +$$\cdots\rtarr E_q(Y,C)\overto{\ps} E_q(Y,A)\oplus E_q(Y,B)\overto{\ph} E_q(Y,X)\overto{\DE} +E_{q-1}(Y,C)\rtarr \cdots.$$ +Here, if $i: (Y,C)\rtarr (Y,A)$, $j: (Y,C)\rtarr (Y,B)$, $k: (Y,A)\rtarr (Y,X)$, and +$\ell: (Y,B)\rtarr (Y,X)$ are the inclusions, then +$$\psi(c)=(i_*(c),j_*(c)), \ \ \ \ \ph(a,b)= k_*(a)-\ell_*(b),$$ +and $\DE$ is the composite +$$E_q(Y,X)\overto{\pa} E_{q-1}(X,B)\iso E_{q-1}(A,C)\rtarr E_{q-1}(Y,C).$$ +\end{thm} +\begin{proof} +This too is left as an exercise, but it is formally the same exercise. +The relevant diagram is the following one: +$$\diagram +& & E_q(Y,C) \dlto \ddto \drto & & \\ +& E_q(Y,B) \drto & & E_q(Y,A) \dlto & \\ +& & E_q(Y,X) \dlto_{\pa} \ddto^{\pa} \drto^{\pa} \xto '[0,2] '[4,2]^{\DE} '[4,1] +\xto '[0,-2] '[4,-2]_{-\DE} '[4,-1] & & \\ +& E_{q-1}(X,A) & & E_{q-1}(X,B) & \\ +& & E_{q-1}(X,C) \ulto \ddto \urto & & \\ +& E_{q-1}(B,C) \drto \urto \uuto^{\iso} & & E_{q-1}(A,C) \dlto \ulto +\uuto_{\iso} & \\ +& & E_{q-1}(Y,C) & & \\ +\enddiagram$$ +Alternatively, one can use CW approximation. For a CW triad $(X;A,B)$, with +$X$ a subcomplex of a CW complex $Y$, there is a short exact sequence +$$0\rtarr C_*(Y/C)\rtarr C_*(Y/A)\oplus C_*(Y/B)\rtarr C_*(Y/X)\rtarr 0$$ +whose associated long exact sequence is the relative Mayer-Vietoris sequence. +\end{proof} + +A comparison of definitions gives a relationship between these sequences. + +\begin{cor} +The absolute and relative Mayer-Vietoris sequences are related by the following +commutative diagram: +$$\diagram +E_q(Y,C)\rto^(0.35){\ps} \dto_{\pa} & E_q(Y,A)\oplus E_q(Y,B) \rto^(0.6){\ph} +\dto^{\pa+\pa} & E_q(Y,X) \rto^{\DE} \dto^{\pa} +& E_{q-1}(Y,C) \dto^{\pa}\\ +E_{q-1}(C)\rto_(0.35){\ps} & + E_{q-1}(A)\oplus E_{q-1}(B)\rto_(0.6){\ph} & E_{q-1}(X)\rto_{\DE} +& E_{q-2}(C).\\ +\enddiagram$$ +\end{cor} + +\section{The homology of colimits} + +In this section, we let $X$ be the union of an expanding sequence of subspaces $X_i$, +$i\geq 0$. We have seen that the compactness of spheres $S^n$ and cylinders $S^n\times I$ +implies that, for any choice of basepoint in $X_0$, the natural map +$$\colim\,\pi_*(X_i)\rtarr \pi_*(X)$$ +is an isomorphism. We shall use the additivity and weak equivalence axioms and the +Mayer-Vietoris sequence to prove the analogue for homology. + +\begin{thm} +The natural map +$$\colim E_*(X_i)\rtarr E_*(X)$$ +is an isomorphism.\index{colimit!homology of} +\end{thm} + +We record an algebraic description of the colimit of a sequence for use in the proof. + +\begin{lem} +Let $f_i: A_i\rtarr A_{i+1}$ be a sequence of homomorphisms of Abelian groups. +Then there is a short exact sequence +$$ 0\rtarr \textstyle{\sum}_i A_i\overto{\al} \textstyle{\sum}_i A_i\overto{\be} \colim\,A_i\rtarr 0,$$ +where $\al(a_i)=a_i-f_i(a_i)$ for $a_i\in A_i$ and the restriction of $\be$ to $A_i$ +is the canonical map given by the definition of a colimit. +\end{lem} + +By the additivity axiom, we may as well assume that $X$ and the $X_i$ are path connected. +The proof makes use of a useful general construction called the ``telescope'' of the $X_i$, +denoted $\tel\,X_i$. Let $j_i: X_i\rtarr X_{i+1}$ be the given inclusions and consider the +mapping cylinders +$$M_{i+1}=(X_i\times[i,i+1])\cup X_{i+1}$$ +that are obtained by identifying $(x,i+1)$ with $j_i(x)$ for $x\in X_i$. +Inductively, let $Y_0=X_0\times\sset{0}$ and suppose that we have constructed +$Y_i\supset X_i\times \sset{i}$. Define $Y_{i+1}$ to be the double mapping cylinder +$Y_i\cup M_{i+1}$ obtained by identifying $(x,i)\in Y_i$ with $(x,i)\in M_{i+1}$ for +$x\in X_i$. Define $\tel\,X_i$\index{telescope} to be the union of the $Y_i$, +with the colimit topology. Thus +$$\tel\,X_i= \bigcup_i X_i\times [i,i+1],$$ +with the evident identifications at the ends of the cylinders. + +Using the retractions of the mapping cylinders, we obtain composite retractions +$r_i: Y_i\rtarr X_i$ such that the following diagrams commute +$$\diagram +Y_i\rto^{\subset} \dto_{r_i} & Y_{i+1} \dto^{r_{i+1}} \\ +X_i \rto_{j_i} & X_{i+1} \\ +\enddiagram$$ +Since the $r_i$ are homotopy equivalences and since homotopy groups commute with colimits, it +follows that we obtain a weak equivalence +$$r: \tel X_i\rtarr X$$ +on passage to colimits. By the weak equivalence axiom, $r$ induces an isomorphism on +homology. It therefore suffices to prove that the natural map +$$\colim\,E_*(X_i)\iso \colim\,E_*(Y_i)\rtarr E_*(\tel\,X_i)$$ +is an isomorphism. We define subspaces $A$ and $B$ of $\tel\,X_i$ by choosing $\epz<1$ and +letting +$$A =X_0\times[0,1]\, \textstyle{\coprod} \, +\textstyle{\coprod}_{i\geq 1}\, X_{2i-1}\times[2i-\epz,2i] +\cup X_{2i}\times [2i,2i+1]$$ +and +$$B=\textstyle{\coprod}_{i\geq 0}\,X_{2i}\times[2i+1-\epz,2i+1] +\cup X_{2i+1}\times [2i+1,2i+2].$$ +We let $C=A\cap B$ and find that +$$C=\textstyle{\coprod}_{i\geq 0}\, X_i\times [i+1-\epz,i+1].$$ +This gives an excisive triad, and a quick inspection shows that we have canonical +homotopy equivalences +$$A\htp \textstyle{\coprod}_{i\geq 0} X_{2i},\ \ B\htp\textstyle{\coprod}_{i\geq 0}X_{2i+1},\ \ \tand +C\htp\textstyle{\coprod}_{i\geq 0}X_i.$$ +Moreover, under these equivalences the inclusion $C\rtarr A$ has restrictions +$$\id: X_{2i}\rtarr X_{2i} \ \ \tand\ \ j_{2i+1}: X_{2i+1}\rtarr X_{2i+2},$$ +while the inclusion $C\rtarr B$ has restrictions +$$j_{2i}: X_{2i}\rtarr X_{2i+1} \ \ \tand\ \ \id: X_{2i+1}\rtarr X_{2i+1}.$$ +By the additivity axiom, +$$E_*(A)=\textstyle{\sum}_i E_*(X_{2i}),\ \ E_*(B)=\textstyle{\sum}_i E_*(X_{2i+1}), +\ \tand\ E_*(C)=\textstyle{\sum}_i E_*(X_i).$$ +We construct the following commutative diagram, whose top row is the Mayer-Vietoris +sequence of the triad $(\text{tel}\, X_i;A,B)$ and whose bottom row is a short exact sequence +as displayed in our algebraic description of colimits: +$$\diagram +\cdots \rto & E_q(C) \rto \dto_{\iso} & E_q(A)\oplus E_q(B) \rto \dto^{\iso} + & E_q(\tel X_i) \rto \dto^{\iso} & \cdots \\ +\cdots \rto & \sum_i E_q(X_i) \rto^{\al'} \dto_{\sum(-1)^{i}} + & \sum_i E_q(X_i) \rto^{\be'} \dto^{\sum_i(-1)^{i}} +& E_q(X) \rto \ddashed^{\xi}|>\tip & \cdots \\ +0 \rto & \sum_i E_q(X_i) \rto^{\al} + & \sum_i E_q(X_i) \rto^{\be} +& \colim E_q(X_i) \rto & 0. \\ +\enddiagram$$ +By the definition of the maps in the Mayer-Vietoris sequence, $\al'(x_i)=x_i+(j_i)_*(x_i)$ +and $\be'_i(x_i)=(-1)^i(k_i)_*(x_i)$ for $x_i\in E_q(X_i)$, where $k_i: X_i\rtarr X$ is +the inclusion. The commutativity of the lower left square is just the relation +$$(\textstyle{\sum}_i (-1)^i)\al'(x_i)=(-1)^i(x_i-(j_i)_*(x_i)).$$ +The diagram implies the required isomorphism $\xi$. + +\begin{rem} There is a general theory of ``homotopy colimits,''\index{homotopy colimit} which are +up to homotopy versions of +colimits. The telescope is the homotopy colimit of a sequence. The double mapping cylinder +that we used in approximating excisive triads by CW triads is the homotopy pushout of a +diagram of the shape $\bullet\longleftarrow \bullet\rtarr \bullet$. We implicitly used +homotopy coequalizers in constructing CW approximations of spaces. +\end{rem} + +\vspace{.1in} + +\begin{center} +PROBLEM +\end{center} +\begin{enumerate} +\item Complete the proof that the Mayer-Vietoris sequence is exact. +\end{enumerate} + +\chapter{The Hurewicz and uniqueness theorems} + +We now return to the context of ``ordinary homology theories,''\index{homology theory!ordinary} +namely those that satisfy the dimension axiom. We prove a fundamental relationship, called the Hurewicz theorem, +between homotopy groups and homology groups. We then use it to prove the uniqueness of ordinary +homology with coefficients in $\pi$. + +\section{The Hurewicz theorem} + +Although the reader may prefer to think in terms of the cellular homology theory already +constructed, the proof of the Hurewicz theorem depends only on the axioms. It is this fact +that will allow us to use the result to prove the uniqueness of homology theories in the +next section. We take $\pi=\bZ$ and delete it from the notation. The dimension axiom +implicitly fixes a generator $i_0$ of $\tilde{H}_0(S^0)$, and we choose generators $i_n$ of +$\tilde{H}_n(S^n)$ inductively by setting $\SI i_n=i_{n+1}$. + +\begin{defn} +For based spaces $X$, define the Hurewicz homomorphism\index{Hurewicz homomorphism} +$$h:\pi_n(X)\rtarr \tilde{H}_n(X)$$ +by +$$h([f])=f_*(i_n).$$ +\end{defn} + +\begin{lem} If $n\geq 1$, then $h$ is a homomorphism for all $X$. +\end{lem} +\begin{proof} +For maps $f,g: S^n\rtarr X$, $[f+g]$ is represented by the composite +$$S^n\overto{p} S^n\wed S^n\overto{f\wed g} X\wed X\overto{\triangledown} X,$$ +where $p$ is the pinch map and $\triangledown$ is the codiagonal map; that is, +$\triangledown$ restricts to the identity on each wedge summand. Since +$p_*(i_n)=i_n+i_n$ and $\triangledown$ induces addition on $\tilde{H}_*(X)$, +the conclusion follows. +\end{proof} + +\begin{lem} The Hurewicz homomorphism is natural and the following diagram +commutes for $n\geq 0$: +$$\diagram +\pi_n(X) \dto_{\SI} \rto & \tilde{H}_n(X) \dto^{\SI} \\ +\pi_{n+1}(\SI X) \rto_{h} & \tilde{H}_{n+1}(\SI X).\\ +\enddiagram$$ +\end{lem} +\begin{proof} +The naturality of $h$ is clear, and the naturality of $\SI$ on homology +implies the commutativity of the diagram: +$$(h\com \SI)([f])=(\SI f)_*(\SI i_n)=\SI(f_*(i_n))=\SI(h([f])). \qed $$ +\renewcommand{\qed}{}\end{proof} + +\begin{lem} Let $X$ be a wedge of $n$-spheres. Then +$$h:\pi_n(X)\rtarr \tilde{H}_n(X)$$ +is the Abelianization homomorphism if $n=1$ and is an isomorphism if $n>1$. +\end{lem} +\begin{proof} +When $X$ is a single sphere, $h[\id]=i_n$ and the conclusion is obvious. +In general, $\pi_n(X)$ is the free group if $n=1$ or the free Abelian group +if $n\geq 2$ with generators given by the inclusions of the wedge summands. +Since $h$ maps these generators to the canonical generators of +the free Abelian group $\tilde{H}_n(X)$, the conclusion follows. +\end{proof} + +That is all that we shall need in the next section, but we can generalize +the lemma to arbitrary $(n-1)$-connected based spaces $X$. + +\begin{thm}[Hurewicz]\index{Hurewicz theorem} +Let $X$ be any $(n-1)$-connected based space. Then +$$h:\pi_n(X)\rtarr \tilde{H}_n(X)$$ +is the Abelianization homomorphism if $n=1$ and is an isomorphism if $n>1$. +\end{thm} +\begin{proof} +We can assume without loss of generality that $X$ is a CW complex with a single +vertex, based attaching maps, and no $q$-cells for $1\leq q1$, the homotopy excision theorem +implies that the top row is exact. To see this, factor $f$ as the composite of +the inclusion $K\rtarr Mf$ and the deformation retraction $r:Mf\rtarr L$. Since +$X=Cf$, we have the following commutative diagram, in which the top row is exact: +$$\diagram +\pi_n(K) \rto \ddouble & \pi_n(Mf)\rto \dto^{r_*} & \pi_n(Mf,K)\rto \dto & 0 \\ +\pi_n(K) \rto & \pi_n(L) \rto & \pi_n(X) \rto & 0.\\ +\enddiagram$$ +Since $K$ and $L$ are $(n-1)$-connected and $n>1$, a corollary of the homotopy +excision theorem gives that $X$ is $(n-1)$-connected and +$\pi_n(Mf,K)\rtarr \pi_n(X)$ is an isomorphism. +\end{proof} + +\section{The uniqueness of the homology of CW complexes} + +We assume given an ordinary homology theory on CW pairs and describe how it must +be computed. We focus on integral homology, taking $\pi=\bZ$ and deleting it from +the notation. With a moment's reflection on the case $n=0$, we see that the +Hurewicz theorem gives a natural isomorphism\index{reduced homology!provisional definition} +$$\tilde{H}'_n(X)\rtarr \tilde{H}_n(X)$$ +for $(n-1)$-connected based spaces $X$. Here the groups on the left are defined +in terms of homotopy groups and were used in our construction of cellular chains, +while the groups on the right are those of our given homology theory. We use the +groups on the right to construct cellular chains in our given theory, and we find +that the isomorphism is compatible with differentials. From here, to prove uniqueness, +we only need to check from the axioms that our given theory is computable from the +homology groups of these cellular chain complexes. + +Thus let $X$ be a CW complex. For each integer $n$, define +$$C_n(X)=H_n(X^n,X^{n-1})\iso\tilde{H}_n(X^n/X^{n-1}).$$ +Define +$$d: C_n(X)\rtarr C_{n-1}(X)$$ +to be the composite +$$H_n(X^n,X^{n-1})\overto{\pa}H_{n-1}(X^{n-1}) +\rtarr H_{n-1}(X^{n-1},X^{n-2}).$$ +It is not hard to check that $d\com d=0$. +\begin{thm} +$C_*(X)$ is isomorphic to the cellular chain complex of $X$.\index{cellular chain complex} +\end{thm} +\begin{proof} +Since $X^n/X^{n-1}$ is the wedge of an $n$-sphere for each $n$-cell of $X$, we +see by the additivity axiom that $C_n(X)$ is the free Abelian group with one +generator $[j]$ for each $n$-cell $j$. We must compare the differential with +the one that we defined earlier. Let $i:X^{n-1}\rtarr X^n$ be the inclusion. +We see from our proof of the suspension isomorphism that $d$ coincides with the +composite +$$\tilde{H}_n(X^n/X^{n-1})\iso \tilde{H}_n(Ci) +\to \tilde{H}_n(\SI X^{n-1}) +\overto{\SI^{-1}}\tilde{H}_{n-1}(X^{n-1})\to \tilde{H}_{n-1}(X^{n-1}/X^{n-2}).$$ +By the naturality of the Hurewicz homomorphism and its commutation with suspension, +this coincides with the differential that we defined originally. +\end{proof} + +Similarly, if we start with a homology theory $H_*(-;\pi)$, we can use the axioms to +construct a chain complex $C_*(X;\pi)$, and a comparison of definitions then gives an +isomorphism of chain complexes +$$C_*(X;\pi)\iso C_*(X)\otimes \pi.$$ +We have identified our axiomatically derived chain complex of $X$ with the cellular +chain complex of $X$, and we again adopt the notation $C_*(X,A)=\tilde{C}_*(X/A)$. + +\begin{thm}\index{homology theory!ordinary} +There is a natural isomorphism +$$H_*(X,A)\iso H_*(C_*(X,A))$$ +under which the natural transformation $\pa$ agrees with the natural transformation +induced by the connecting homomorphisms associated to the short exact sequences +$$0\rtarr C_*(A)\rtarr C_*(X)\rtarr C_*(X,A)\rtarr 0.$$ +\end{thm} +\begin{proof} +In view of our comparison of theories on pairs of spaces and theories on pairs of +CW complexes and our comparison of theories on pairs with reduced theories, it +suffices to obtain a natural isomorphism of reduced theories on based CW complexes $X$. +By the additivity axiom, we may as well assume that $X$ is connected. More precisely, +we must obtain a system of natural isomorphisms +$$\tilde{H}_n(X)\iso {H}_*(\tilde{C}_n(X))$$ +that commute with the suspension isomorphisms. + +By the dimension and additivity axioms, we know the homology of wedges of spheres. +Since $X^n/X^{n-1}$ is a wedge of $n$-spheres, +the long exact homology sequence associated to the cofiber sequence +$$X^{n-1}\rtarr X^n\rtarr X^n/X^{n-1}$$ +and an induction on $n$ imply that +$$\tilde{H}_q(X^{n-1})\rtarr \tilde{H}_q(X^{n})$$ +is an isomorphism for $qn$. Of course, the analogues for cellular homology are obvious. Note in particular +that $\tilde{H}_n(X^{n+1})\iso \tilde{H}_n(X^{n+i})$ for all $i>1$. Since homology +commutes with colimits on sequences of inclusions, this implies that the inclusion +$X^{n+1}\rtarr X$ induces an isomorphism +$$ \tilde{H}_n(X^{n+1})\rtarr \tilde{H}_n(X).$$ +Using these facts, we easily check from the exactness axiom that the rows and columns are +exact in the following commutative diagram: +$$\diagram +& \tilde{H}_{n+1}(X^{n+1}/X^n) \drto^{d_{n+1}} \dto_{\pa} & & 0 \dto \\ +0 \rto & \tilde{H}_n(X^n) \rto_{\rh_*} \dto_{i_*} & \tilde{H}_n(X^n/X^{n-1}) +\rto^{\pa} \drto_{d_n} & \tilde{H}_{n-1}(X^{n-1}) \dto \\ +& \tilde{H}_n(X)\iso\tilde{H}_n(X^{n+1}) \dto & & \tilde{H}_{n-1}(X^{n-1}/X^{n-2}). \\ +& 0 & & \\ +\enddiagram$$ +Define $\al: \tilde{H}_n(X)\rtarr H_n(\tilde{C}_*(X))$ by letting $\al(x)$ be the +homology class of $\rh_*(y)$ for any $y$ such that $i_*(y)=x$. It is an exercise in +diagram chasing and the definition of the homology of a chain complex to check that +$\al$ is a well defined isomorphism. + +The reduced chain complex of $\SI X$ can be +identified with the suspension of the reduced chain complex of $X$. That is, +$$\tilde{C}_{n+1}(\SI X)\iso \tilde{C}_n(X),$$ +compatibly with the differential. All maps in the diagram commute with suspension, +and this implies that the isomorphisms $\al$ commute with the suspension isomorphisms. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Let $\pi$ be any group. Construct a connected CW complex $K(\pi,1)$ such that +$\pi_1(K(\pi,1))=\pi$ and $\pi_q(K(\pi,1))=0$ for $q\neq 1$. +\item* In Problem 1, it is rarely the case that $K(\pi,1)$ can be constructed as a compact +manifold. What is a necessary condition on $\pi$ for this to happen? +\item Let $n\geq1$ and let $\pi$ be an Abelian group. Construct a connected CW complex +$M(\pi,n)$ such that $\tilde{H}_n(X;\bZ)=\pi$ and $\tilde{H}_q(X;\bZ)=0$ for $q\neq n$. +(Hint: construct $M(\pi,n)$ as the cofiber of a map between wedges of spheres.) +The spaces $M(\pi,n)$ are called Moore spaces.\index{Moore space} +\item Let $n\geq1$ and let $\pi$ be an Abelian group. Construct a connected CW complex +$K(\pi,n)$ such that $\pi_n(X)=\pi$ and $\pi_q(X)=0$ for $q\neq n$. (Hint: +start with $M(\pi,n)$, using the Hurewicz theorem, and kill its higher homotopy groups.) +The spaces $K(\pi,n)$ are called Eilenberg-Mac\,Lane spaces.\index{Eilenberg-Mac\,Lane space} +\item There are familiar spaces that give $K(\bZ,1)$, $K(\bZ_2,1)$, and $K(\bZ,2)$. Name them. +\item Let $X$ be any connected CW complex whose only non-vanishing homotopy +group is $\pi_n(X)\iso \pi$. Construct a homotopy equivalence $K(\pi,n)\rtarr X$, +where $K(\pi,n)$ is the Eilenberg-Mac\,Lane space that you have constructed. +\item* For groups $\pi$ and $\rh$, compute $[K(\pi,n),K(\rh,n)]$; here $[-,-]$ +means based homotopy classes of based maps. +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{Singular homology theory} + +We explain, without giving full details, how the standard approach to singular homology +theory fits into our framework. We also introduce simplicial sets and spaces and their +geometric realization. These notions play a fundamental role in modern algebraic topology. + +\section{The singular chain complex} + +The standard topological $n$-simplex\index{nsimplex@$n$-simplex!topological} is the subspace +$$\DE_n=\sset{(t_0,\ldots\!,t_n)|0\leq t_i\leq 1,\ \textstyle{\sum} t_i=1}$$ +of $\bR^{n+1}$. There are ``face maps''\index{face map} +$$\de_i: \DE_{n-1}\rtarr \DE_n,\ \ 0\leq i\leq n,$$ +specified by +$$\de_i(t_0,\ldots\!,t_{n-1})=(t_0,\ldots\!,t_{i-1},0,t_i,\ldots\!,t_{n-1})$$ +and ``degeneracy maps''\index{degeneracy map} +$$\si_i: \DE_{n+1}\rtarr \DE_{n},\ \ 0\leq i\leq n,$$ +specified by +$$\si_i(t_0,\ldots\!,t_{n+1})=(t_0,\ldots\!,t_{i-1},t_i+t_{i+1},\ldots\!, t_{n+1}).$$ + +For a space $X$, define $S_nX$ to be the set of continuous maps $f:\DE_n\rtarr X$. +In particular, regarding a point of $X$ as the map that sends $1$ to $x$, we may +identify the underlying set of $X$ with $S_0X$. Define the $i$th face operator +$$d_i: S_nX\rtarr S_{n-1}X,\ \ 0\leq i\leq n,$$ +by +$$d_i(f)(u)=f(\de_i(u)),$$ +where $u\in \DE_{n-1}$, and define the $i$th degeneracy operator +$$s_i: S_nX\rtarr S_{n+1}X, \ \ 0\leq i\leq n,$$ +by +$$s_i(f)(v)=f(\si_i(v)),$$ +where $v\in \DE_{n+1}$. The following identities are easily checked: +$$d_i\com d_j=d_{j-1}\com d_i \ \ \ \text{if}\ \ ij+1. +\end{cases}$$ +$$s_i\com s_j=s_{j+1}\com s_i \ \ \text{if}\ \ i\leq j.$$ +A map $f: \DE_n\rtarr X$ is called a singular $n$-simplex.\index{nsimplex@$n$-simplex!singular} It +is said to be nondegenerate +if it is not of the form $s_i(g)$ for any $i$ and $g$. Let $C_n(X)$\index{singular chain +complex} be the free Abelian +group generated by the nondegenerate $n$-simplexes, and think of $C_n(X)$ as the quotient +of the free Abelian group generated by all singular $n$-simplexes by the subgroup generated +by the degenerate $n$-simplexes. Define +$$d = \sum_{i=0}^{n}(-1)^i d_i: C_n(X)\rtarr C_{n-1}(X).$$ +The identities ensure that $C_*(X)$ is then a well defined chain complex. In fact, +$$d\com d =\sum_{i=0}^{n-1}\sum _{j=0}^{n}(-1)^{i+j}d_i\com d_j,$$ +and, for $i^{d_n} & B_{n-1} \ldashed<.5ex>^{\si_n}|>\tip \rto & 0; \\ +\enddiagram$$ +we choose a splitting $\si_n$ of the second. Writing $f^*=\Hom(f,M)$ consistently, we obtain a +commutative diagram with exact rows and columns +$$\diagram + & & 0 & 0 \dto & \\ +0 \rto & \Hom(H_n,M) \rto^{\pi_n^*} & \Hom(Z_n,M) \rto^{i_n^*} \uto +& \Hom(B_n,M) \dto^{d_{n+1}^*} & \\ +\cdots \rto & \Hom(X_{n-1},M) \rto^{d_n^*} \dto_{j_{n-1}^*} & \Hom(X_n,M) \uto_{j_n^*} +\rto^{d_{n+1}^*} \ddashed<-.5ex>_{\si_n^*}|>\tip & \Hom(X_{n+1},M) \rto & \cdots \\ + & \Hom(Z_{n-1},M) \rto_{i_{n-1}^*} \dto +& \Hom(B_{n-1},M) \rto^{\de} \uto<-.5ex>_{d_n^*} \urdashed_{0}|>\tip & \Ext^1_R(H_{n-1},M)\rto & 0 \\ + & 0 & 0 \uto & & \\ +\enddiagram$$ +By inspection of the diagram, we see that the canonical map $\al$ coincides with the composite +$$H^n(X;M)=\ker\,d^*_{n+1}/\im\,d^*_n = \ker\,i^*_nj^*_n/\im\,d_n^*i^*_{n-1} \overto{j_n^*} + \im \pi^*_n\,\overto{(\pi_n^*)^{-1}} \Hom(H_n,M).$$ +Since $j_n^*$ is an epimorphism, so is $\al$. The kernel of $\al$ is +$\im\,d_n^*/\im\,d_n^*i^*_{n-1}$, and $\de(d_n^*)^{-1}$ maps this group isomorphically onto +$\Ext^1_R(H_{n-1},M)$. The composite $\de\si_n^*$ induces the required splitting. +\end{proof} + +\section{Relations between $\otimes$ and Hom} + +We shall need some observations about cochain complexes and tensor products, and +we first recall some general facts about the category of $R$-modules. For $R$-modules $L$, $M$, +and $N$, we have an adjunction +$$\Hom(L\ten M,N)\iso \Hom(L,\Hom(M,N)).$$ +We also have a natural homomorphism +$$\Hom(L,M)\ten N\rtarr \Hom(L,M\ten N),$$ +and this is an isomorphism if either $L$ or $N$ is a finitely generated projective $R$-module. +Again, we have a natural map +$$\Hom(L,M)\ten \Hom(L',M')\rtarr \Hom(L\ten L',M\ten M'),$$ +which is an isomorphism if $L$ and $L'$ are finitely generated and projective or if $L$ is +finitely generated and projective and $M=R$. + +We can replace $L$ and $L'$ by chain complexes and obtain similar maps, inserting signs +where needed. For example, a chain homotopy $X\ten \sI\rtarr X'$ between chain maps +$f,g: X\rtarr X'$ induces a chain map +$$\Hom(X',M)\rtarr \Hom (X\ten \sI,M)\iso \Hom(\sI,\Hom(X,M))\iso \Hom(X,M)\ten\sI^*,$$ +where $\sI^*=\Hom(\sI,R)$. It should be clear that this implies that our original chain +homotopy induces a homotopy of cochain maps +$$f^*\htp g^*: \Hom(X',M)\rtarr \Hom(X,M).$$ + +If $Y$ and $Y'$ are cochain complexes, then we have the natural homomorphism +$$\al: H^*(Y)\ten H^*(Y')\rtarr H^*(Y\ten Y')$$ +given by $\al([y]\ten [y'])=[y\ten y']$, exactly as for chain complexes. (In fact, by +regrading, we may view this as a special case of the map for chain complexes.) The K\"{u}nneth +theorem applies to this map. For its flatness hypothesis, it is useful to remember that, for +any Noetherian ring $R$, the dual $\Hom(F,R)$ of a free $R$-module is a flat $R$-module. + +As indicated above, if $Y=\Hom(X,M)$ and $Y'=\Hom(X',M')$ for chain complexes $X$ and $X'$ +and $R$-modules $M$ and $M'$, then we also have the map of cochain complexes +$$\om: \Hom(X,M)\ten \Hom(X',M')\rtarr \Hom(X\ten X',M\ten M')$$ +specified by the formula +$$\om(f\ten f')(x\ten x') = (-1)^{(\text{deg}\,f')(\text{deg}\, x)}f(x)\ten f'(x').$$ +We continue to write $\om$ for the map it induces on cohomology, and we then have the composite +$$\om\com\al: H^*(X;M)\ten H^*(X';M')\rtarr H^*(X\ten X';M\ten M').$$ +When $M=M'=A$ is a commutative $R$-algebra, we may compose with the map +$$ H^*(X\ten X';A\ten A) \rtarr H^*(X\ten X';A)$$ +induced by the multiplication of $A$ to obtain a map +$$H^*(X;A)\ten H^*(X';A)\rtarr H^*(X\ten X';A).$$ +We are especially interested in the case when $R=\bZ$ and $A$ is either $\bZ$ or a field. + +\chapter{Axiomatic and cellular cohomology theory} + +We give a treatment of cohomology that is precisely parallel to our treatment of +homology. The essential new feature is the cup product structure that makes the +cohomology of $X$ with coefficients in a commutative ring $R$ a commutative graded +$R$-algebra. This additional structure ties together the cohomology groups in +different degrees and is fundamentally important to most of the applications. + +\section{Axioms for cohomology} + +Fix an Abelian group $\pi$ and consider pairs of spaces $(X,A)$. We +shall see that $\pi$ determines a ``cohomology theory on pairs $(X,A)$.''\index{cohomology +theory} + +\begin{thm} For integers $q$, there exist {\em contravariant} functors $H^q(X,A;\pi)$ +from the homotopy category of pairs of spaces to the category of Abelian groups together +with natural transformations $\de: H^q(A;\pi)\rtarr H^{q+1}(X,A;\pi)$, where +$H^q(X;\pi)$ is defined to be $H^q(X,\emptyset;\pi)$. These functors and natural +transformations satisfy and are characterized by the following axioms. +\begin{itemize} +\item DIMENSION\index{dimension axiom}\ \ If $X$ is a point, then $H^0(X;\pi) = \pi$ and $H^q(X;\pi)=0$ +for all other integers. +\item EXACTNESS\index{exactness axiom}\ \ The following sequence is exact, where the unlabeled arrows +are induced by the inclusions $A\rtarr X$ and $(X,\emptyset)\rtarr (X,A)$: +$$\cdots\rtarr H^q(X,A;\pi)\rtarr H^q(X;\pi)\rtarr +H^q(A;\pi)\overto{\de} H^{q+1}(X,A;\pi)\rtarr \cdots .$$ +\item EXCISION\index{excision axiom}\ \ +If $(X;A,B)$ is an excisive triad, so that $X$ is the union of the interiors +of $A$ and $B$, then the inclusion $(A,A\cap B)\rtarr (X,B)$ induces an +isomorphism +$$H^*(X,B;\pi)\rtarr H^*(A,A\cap B;\pi).$$ +\item ADDITIVITY\index{additivity axiom}\ \ +If $(X,A)$ is the disjoint union of a set of pairs $(X_i,A_i)$, then +the inclusions $(X_i,A_i)\rtarr (X,A)$ induce an isomorphism +$$ H^*(X,A;\pi)\rtarr \textstyle{\prod}_i\, H^*(X_i,A_i;\pi).$$ +\item WEAK EQUIVALENCE\index{weak equivalence axiom}\ \ If $f:(X,A)\rtarr (Y,B)$ is a weak +equivalence, then +$$f^*: H^*(Y,B;\pi)\rtarr H^*(X,A;\pi)$$ +is an isomorphism. +\end{itemize} +\end{thm} + +We write $f^*$ instead of $H^*(f)$ or $H^q(f)$. As in homology, our approximation theorems +for spaces, pairs, maps, homotopies, and excisive triads directly imply that such a theory +determines and is determined by an appropriate theory defined on CW pairs, as spelled out +in the following CW version of the theorem.\index{cohomology theory} + +\begin{thm} For integers $q$, there exist functors $H^q(X,A;\pi)$ from the +homotopy category of pairs of CW complexes to the category of +Abelian groups together with natural transformations $\de: H^q(A)\rtarr H^{q+1}(X,A;\pi)$, +where $H^q(X;\pi)$ is defined to be $H^q(X,\emptyset;\pi)$. These functors and natural +transformations satisfy and are characterized by the following axioms. +\begin{itemize} +\item DIMENSION\index{dimension axiom}\ \ If $X$ is a point, then $H^0(X;\pi) = \pi$ and $H^q(X;\pi)=0$ +for all other integers. +\item EXACTNESS\index{exactness axiom}\ \ The following sequence is exact, where the unlabeled arrows +are induced by the inclusions $A\rtarr X$ and $(X,\emptyset)\rtarr (X,A)$: +$$\cdots\rtarr H^q(X,A;\pi)\rtarr H^q(X;\pi)\rtarr +H^q(A;\pi)\overto{\de} H^{q+1}(X,A;\pi)\rtarr \cdots .$$ +\item EXCISION\index{excision axiom}\ \ +If $X$ is the union of subcomplexes $A$ and $B$, then the inclusion +$(A,A\cap B)\rtarr (X,B)$ induces an isomorphism +$$ H^*(X,B;\pi) \rtarr H^*(A,A\cap B;\pi).$$ +\item ADDITIVITY\index{additivity axiom}\ \ +If $(X,A)$ is the disjoint union of a set of pairs $(X_i,A_i)$, then +the inclusions $(X_i,A_i)\rtarr (X,A)$ induce an isomorphism +$$H^*(X,A;\pi)\rtarr \textstyle{\prod}_i\, H^*(X_i,A_i;\pi).$$ +\end{itemize} +Such a theory determines and is determined by a theory as in the previous +theorem. +\end{thm} + +\section{Cellular and singular cohomology} + +We define the cellular cochains\index{cellular cochains} of a CW pair $(X,A)$ with +coefficients in an Abelian group $\pi$ to be +$$C^*(X,A;\pi)=\Hom(C_*(X,A),\pi).$$ +We then define the cellular cohomology groups to be +$$H^*(X,A;\pi)=H^*(C^*(X,A;\pi)).$$ +If $M$ is a module over a commutative ring $R$, we have a natural identification +$$C^*(X,A;M)\iso \Hom_R(C_*(X,A)\ten R,M)$$ +which allows us to do homological algebra over $R$ rather than over $\bZ$ when +convenient. In particular, if $R$ is a field, then +$$ H^*(X,A;M)\iso \Hom_R(H_*(X,A;R),M).$$ +In general, with $R=\bZ$, we have a natural and splittable short exact sequence +$$0\rtarr \Ext^1_{\bZ}(H_{n-1}(X,A),\pi)\rtarr H^n(X,A;\pi)\rtarr \Hom(H_n(X,A),\pi)\rtarr 0.$$ + +The verification of the axioms listed in the previous section is immediate, as in homology. The +fact that cellularly homotopic maps induce the same map on cohomology uses our observations +relating homotopies of chain complexes with homotopies of cochain complexes. For exactness, the +fact that our chain complexes are free over $\bZ$ implies that we have a short exact sequence of +cochain complexes +$$ 0\rtarr C^*(X,A;\pi)\rtarr C^*(X;\pi)\rtarr C^*(A;\pi) \rtarr 0.$$ +The required natural long exact sequence follows. The rest is the same as in homology. + +For general spaces $X$, we can use $\GA X=|S_*X|$ as a canonical CW approximation functor. +We define the singular cochains\index{singular cochains} of $X$ to be the cellular cochains +of $\GA X$. Then our +passage from the cohomology of CW complexes to the cohomology of general spaces can be +realized by taking the cohomology of singular cochain complexes. + +\section{Cup products in cohomology} + +If $X$ and $Y$ are CW complexes, we have an isomorphism +$$C_*(X\times Y)\iso C_*(X)\ten C_*(Y)$$ +of chain complexes and therefore, for any Abelian groups $\pi$ and $\pi'$, an +isomorphism of cochain complexes +$$C^*(X\times Y;\pi\ten \pi')\iso \Hom(C_*(X)\ten C_*(Y),\pi\ten \pi').$$ +By our observations about cochain complexes, there results a natural homomorphism +$$H^*(X;\pi)\ten H^*(Y;\pi')\rtarr H^*(X\times Y;\pi\ten\pi').$$ +If $X=Y$ and if $\pi=\pi'=R$ is a commutative ring, we can use the +diagonal map $\DE: X\rtarr X\times X$ and the product $R\ten R\rtarr R$ to +obtain a ``cup product''\index{cup product} +$$\cup: H^*(X;R)\ten_R H^*(X;R)\rtarr H^*(X;R).$$ +More precisely, for $p\geq 0$ and $q\geq 0$, we have a product +$$\cup: H^p(X;R)\ten_R H^q(X;R)\rtarr H^{p+q}(X;R).$$ +We have noted that we can use $C_*(X;R)$ instead of $C_*(X)$ and so justify +tensoring over $R$ rather than $\bZ$. This product makes $H^*(X;R)$ into +a graded unital, associative, and ``commutative'' $R$-algebra. Here commutativity +is understood in the appropriate graded sense,\index{commutativity!graded} namely +$$xy=(-1)^{pq} yx \ \ \text{if}\ \ \text{deg}\,x=p\ \tand\ \text{deg}\,y=q.$$ +The image of $1\in R=H^0(*;R)$ under the map $\pi^*:H^0(*;R)\rtarr H^0(X;R)$ +induced by the unique map $\pi: X\rtarr \sset{*}$ is the unit (= identity element) for the product. +In fact, the diagrams that say that $H^*(X;R)$ is unital, associative, and commutative +result by passing to cohomology from the evident commutative diagrams +$$\diagram +& X \dto^{\DE} \drdouble \dldouble & \\ +X\times * & X\times X \lto^{\id\times \pi} \rto_{\pi\times\id} & *\times X, \\ +\enddiagram$$ +$$\diagram +X \rto^{\DE} \dto_{\DE} & X\times X \dto^{\DE\times \id} \\ +X\times X \rto_(0.4){\id\times\DE} & X\times X\times X,\\ +\enddiagram$$ +and +$$\diagram +& X \dlto_{\DE} \drto^{\DE} & \\ +X\times X\rrto_t & & X\times X.\\ +\enddiagram$$ +Here $t: X\times Y\rtarr Y\times X$ is the transposition, $t(x,y)=(y,x)$. The following +diagrams commute in homology and cohomology with cofficients in $R$: +$$\diagram +H_*(X)\ten_{R} H_*(Y)\dto_{\ta} \rto^(0.55){\al} & H_*(X\times Y) \dto^{t_*}\\ +H_*(Y)\ten_{R} H_*(X) \rto^(0.55){\al} & H_*(Y\times X)\\ +\enddiagram$$ +and +$$\diagram +H^*(X)\ten_{R} H^*(Y)\dto_{\ta} \rto^(0.55){\al} & H^*(X\times Y) \dto^{t^*}\\ +H^*(Y)\ten_{R} H^*(X) \rto^(0.55){\al} & H^*(Y\times X). +\enddiagram$$ +In both diagrams, +$$\ta(x\ten y)=(-1)^{pq} y\ten x \ \ \text{if}\ \ \text{deg}\,x=p\ \tand\ \text{deg}\,y=q.$$ +The reason is that, on the topological level, $t$ permutes $p$-cells past $q$-cells and, on +the level of cellular chains, this involves the transposition +$$ S^{p+q} = S^p\sma S^q \rtarr S^q\sma S^p = S^{p+q}.$$ +We leave it as an exercise that this map has degree $(-1)^{pq}$. It is this fact that forces +the cup product to be commutative in the graded sense. + +In principle, the way to compute cup products is to pass to cellular chains from a cellular +approximation to the diagonal map $\DE$. The point is that $\DE$ fails to be cellular since +it carries the $n$-skeleton of $X$ to the $2n$-skeleton of $X\times X$. In practice, this +does not work very well and more indirect means of computation must be used. + +\section{An example: $\bR P^n$ and the Borsuk-Ulam theorem} + +Remember that $\bR P^n$\index{projective space!real} is a CW complex with one $q$-cell for +each $q\leq n$. The differential +on $C_q(\bR P^n)\iso \bZ$ is zero if $q$ is odd and multiplication by $2$ if $q$ is even. +When we dualize to $C^*(\bR P^n)$, we find that the differential on $C^q(\bR P^n)$ is +multiplication by $2$ if $q$ is odd and zero if $q$ is even. We read off that +$$H^q(\bR P^n;\bZ)= +\begin{cases} +\bZ \ \ \ \text{if}\ \ q=0 \\ +\bZ_2 \ \ \text{if}\ \ 0n. +\end{cases}$$ +The reader may find it instructive to compare with the calculations in homology, checking +the correctness of the calculation by comparison with the universal coefficient theorem. + +We shall later use Poincar\'{e} duality to give a quick proof that the cohomology algebra +$H^*(\bR P^n;\bZ_2)$ is a truncated polynomial algebra $\bZ_2[x]/(x^{n+1})$, where $\deg\,x=1$. +That is, for $1\leq q\leq n$, the unique non-zero element of $H^q(\bR P^n;\bZ_2)$ is the +$q$th power of $x$. This means that the elements are so tightly bound together that knowledge +of the cohomological behavior of a map $f: \bR P^m\rtarr \bR P^n$ on cohomology in degree one +determines its behavior on cohomology in all higher degrees. We assume that $m\geq 1$ and +$n\geq 1$ to avoid triviality. + +\begin{prop} Let $f: \bR P^m\rtarr \bR P^n$ be a map such that +$f_*: \pi_1(\bR P^m)\rtarr \pi_1(\bR P^n)$ is non-zero. Then $m\leq n$. +\end{prop} +\begin{proof} +Since $\pi_1(\bR P^1)=\bZ$ and $\pi_1(\bR P^m)=\bZ_2$ if $m\geq 2$, the result is certainly +true if $n=1$. Thus assume that $n>1$ and assume for a contradiction that $m>n$. By the +naturality of the Hurewicz isomorphism, $f_*: H_1(\bR P^m;\bZ)\rtarr H_1(\bR P^n;\bZ)$ is +non-zero. By our universal coefficient theorems, the same is true for mod $2$ homology and +for mod $2$ cohomology. That is, if $x$ is the non-zero element of $H^1(\bR P^n;\bZ_2)$, then +$f^*(x)$ is the non-zero element of $H^1(\bR P^m;\bZ_2)$. +By the naturality of cup products +$$ (f^*(x))^m = f^*(x^m).$$ +However, the left side is non-zero in $H^m(\bR P^m;\bZ_2)$ and the right side is zero since +$x^m=0$ by our assumption that $m>n$. The contradiction establishes the conclusion. +\end{proof} + +We use this fact together with covering space theory to prove a celebrated result +known as the Borsuk-Ulam theorem. A map $g: S^m\rtarr S^n$ is said to be antipodal\index{antipodal +map} if it takes pairs of antipodal points to pairs of antipodal points. It then induces a map +$f: \bR P^m\rtarr \bR P^n$ such that the following diagram commutes: +$$\diagram +S^m\rto^g \dto_{p_m} & S^n \dto^{p_n}\\ +\bR P^m \rto_f & \bR P^n,\\ +\enddiagram$$ +where $p_m$ and $p_n$ are the canonical coverings. + +\begin{thm} If $m>n\geq 1$, then there exist no antipodal maps $S^m\rtarr S^n$. +\end{thm} +\begin{proof} +Suppose given an antipodal map $g:S^m\rtarr S^n$. According to +the proposition, $f_*:\pi_1(\bR P^m)\rtarr \pi_1(\bR P^n)$ is zero. According to +the fundamental theorem of covering space theory, there is a map +$\tilde{f}: \bR P^m\rtarr S^n$ such that $p_n\com\tilde{f}= f$. Let $s\in S^m$. +Then $\tilde{f}(p_m(s))=\tilde{f}(p_m(-s))$ must be either $g(s)$ or $g(-s)$, +since these are the only two points in $p_n^{-1}(f(p_m(s)))$. Thus either $t=s$ +or $t=-s$ satisfies $\tilde{f}(p_m(t))=g(t)$. Therefore, by the fundamental +theorem of covering space theory, the maps $\tilde{f}\com p_m$ and $g$ must be +equal since they agree on a point. This is absurd: $\tilde{f}\com p_m$ takes +antipodal points to the same point, while $g$ was assumed to be antipodal. +\end{proof} + +\begin{thm}[Borsuk-Ulam]\index{Borsuk-Ulam theorem} +For any continuous map $f: S^n\rtarr \bR^n$, there exists $x\in S^n$ such that +$f(x)=f(-x)$. +\end{thm} +\begin{proof} +Suppose for a contradiction that $f(x)\neq f(-x)$ for all $x$. We could then define a +continuous antipodal map $g: S^n\rtarr S^{n-1}$ by letting $g(x)$ be the point at +which the vector from $0$ through $f(x)-f(-x)$ intersects $S^{n-1}$. +\end{proof} + +\section{Obstruction theory} + +We give an outline of one of the most striking features of cohomology: the +cohomology groups of a space $X$ with coefficients in the homotopy groups of a +space $Y$ control the construction of homotopy classes of maps $X\rtarr Y$. +As a matter of motivation, this helps explain why one is interested in general +coefficient groups. It also explains why the letter $\pi$ is so often used to +denote coefficient groups. + +\begin{defn} Fix $n\geq 1$. A connected space $X$ is said to be $n$-simple\index{nsimple +space@$n$-simple space} if +$\pi _{1}(X)$ is Abelian and acts trivially on the homotopy groups $\pi _{q}(X)$ +for $q\leq n$; $X$ is said to be simple\index{simple space} if it is $n$-simple for all $n$. +\end{defn} + +Let $(X,A)$ be a relative CW complex with relative skeleta $X^n$ and let $Y$ be an +$n$-simple space. The +assumption on $Y$ has the effect that we need not worry about basepoints. Let +$f: X^{n}\rtarr Y$ be a map. We ask when $f$ can be extended to +a map $X^{n+1}\rtarr Y$ that restricts to the given map on $A$. + +If we compose the attaching maps $S^{n} \rightarrow X$ of cells of $X\setminus A$ +with $f$, we obtain elements of $\pi_{n}(Y)$. These elements specify a well defined +``obstruction cocycle''\index{obstruction cocycle} +\[ c_{f}\in C^{n+1}(X,A;{\pi}_{n}(Y)). \] +Clearly, by considering extensions cell by cell, $f$ extends to $X^{n+1}$ if and only +if $c_{f} = 0$. This is not a computable +criterion. However, if we allow ourselves to modify $f$ a little, then we can refine the +criterion to a cohomological one that often is computable. If $f$ and $f'$ +are maps $X^{n} \rightarrow Y$ and $h$ is a homotopy rel $A$ of the +restrictions of $f$ and $f'$ to $X^{n-1}$, +then $f$, $f'$, and $h$ together define a map +\[ h(f,f'): (X\times I)^{n} \rtarr Y. \] +Applying $c_{h(f,f')}$ to cells $j\times I$, we obtain a ``deformation +cochain''\index{deformation cochain} +\[ d_{f,f',h}\in C^{n}(X,A;{\pi}_{n}(Y)) \] +such that $\delta d_{f,f',h} = c_{f}-c_{f'}$. Moreover, given $f$ and $d$, +there exists $f'$ that coincides with $f$ on $X^{n-1}$ and satisfies +$d_{f,f'} = d$, where the constant homotopy $h$ is understood. This gives the +following result. + +\begin{thm} +For $f: X^{n}\rtarr Y$, the restriction of $f$ to +$X^{n-1}$ extends to a map $X^{n+1}\rightarrow Y$ if and only if $[c_{f}]=0$ +in $H^{n+1}(X,A;{\pi}_{n}(Y))$. +\end{thm} + +It is natural to ask further when such extensions are unique up to homotopy, +and a similar argument gives the answer. + +\begin{thm} +Given maps $f, f': X^{n}\rightarrow Y$ and a homotopy rel $A$ of their +restrictions to $X^{n-1}$, there is an obstruction class in +$H^{n}(X,A;{\pi}_{n}(Y))$ that vanishes +if and only if the restriction of the given homotopy to $X^{n-2}$ +extends to a homotopy $f\simeq f'$ rel $A$. +\end{thm} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} + +The first few problems here are parallel to those at the end of Chapter 16. +\begin{enumerate} +\item Let $X$ be a space that satisfies the hypotheses used to construct a universal +cover $\tilde{X}$ and let $A$ be an Abelian group. Using cellular or singular chains, +show that +$$C^*(X;A)\iso \Hom_{\bZ[\pi]}(C_*(\tilde{X}),A).$$ +\item Show that there is an isomorphism +$$H^*(K(\pi,1);A) \iso \Ext^*_{\bZ[\pi]}(\bZ,A).$$ +When $A$ is a commutative ring, the Ext groups have algebraically defined products, +constructed as follows. The evident isomorphism $\bZ\iso\bZ\ten\bZ$ is covered by a +map of free $\bZ[\pi]$-resolutions +$P \rtarr P\ten P$, where $\bZ[\pi]$ acts diagonally on tensor products, +$\al(x\ten y) = \al x\ten \al y$. This chain map is unique +up to chain homotopy. It induces a map of chain complexes +$$\Hom_{\bZ[\pi]}(P,A)\ten \Hom_{\bZ[\pi]}(P,A) \rtarr \Hom_{\bZ[\pi]}(P,A)$$ +and therefore an induced product on Ext$^*_{\bZ[\pi]}(\bZ,A)$. +Convince yourself that the isomorphism above preserves +products and explain the intuition (don't worry about technical exactitude). +\item* Now use homological algebra to determine $H^*(\bR P^{\infty};\bZ_2)$ as a ring. +\item Use the previous problem to deduce the ring structure on $H^*(\bR P^n;\bZ_2)$ +for each $n\geq 1$. +\item Let $p: Y\rtarr X$ be a covering space with finite fibers, say of cardinality $n$. +Construct a ``transfer homomorphism''\index{transfer homomorphism} +$t: H^*(Y;A)\rtarr H^*(X;A)$ and show that $t\com p^*: H^*(X;A)\rtarr H^*(X;A)$ +is multiplication by $n$. +\item Let $X$ and $Y$ be CW complexes. Show that the interchange map +$$t: X\times Y\rtarr Y\times X$$ +satisfies $t_*([i]\ten[j])=(-1)^{pq}[j]\ten[i]$ for a $p$-cell of $X$ and a $q$-cell of $Y$. +Deduce that the cohomology ring $H^*(X)$ is commutative in the graded +sense:\index{commutativity!graded} +$$ x\cup y = (-1)^{pq}y\cup x \ \ \text{if}\ \ \text{deg}\,x=p\ \tand\ \text{deg}\,y=q.$$ +\end{enumerate} + +An ``$H$-space''\index{Hspace@$H$-space} is a space $X$ with a basepoint $e$ and a product +$\ph: X\times X\rtarr X$ such that the maps $\la: X\rtarr X$ and $\rh: X\rtarr X$ +given by left and right multiplication by $e$ are each homotopic to the identity map. +Note that $\la$ and $\rh$ specify a map $X\wed X\rtarr X$ that is homotopic to the +codiagonal or folding map $\bigtriangledown$, which restricts to the identity on each +wedge summand. The following two problems are optional review exercises. + +\begin{enumerate} +\item[7.] If $e$ is a nondegenerate basepoint for $X$, then $\ph$ is homotopic to a product +$\ph'$ such that left and right multiplication by $e$ under the product $\ph'$ are +both identity maps. +\item[8.] Show that the product on $\pi_1(X,e)$ induced by the based map +$\ph': X\times X\rtarr X$ agrees with the multiplication given by composition +of paths and that both products are commutative. +\item[9.] For an $H$-space $X$, the following diagram is commutative: +$$\diagram +X\times X \dto_{\ph} \rrto^{\DE\times\DE} & & X\times X\times X\times X +\rrto^{\id\times t\times \id} & & X\times X\times X\times X \dto^{\ph\times \ph} \\ +X \xto[0,4]_{\DE} & & & & X\times X +\enddiagram$$ +(Check it: it is too trivial to write down.) Let $X$ be $(n-1)$-connected, $n\geq 2$, and +let $x\in H^n(X)$. +\begin{enumerate} +\item[(a)] Show that $\ph^*(x) = x\ten 1 + 1\ten x$. +\item[(b)] Show that +$$(\DE\times\DE)^*(\id\times\, t\times \id)^*(\ph\times \ph)^*(x\ten x) +=x^2\ten 1 +(1+(-1)^n)(x\ten x)+1\ten x^2.$$ +\item[(c)] Prove that, if $n$ is even, then either $2(x\ten x)=0$ in $H^*(X\times X)$ or +$x^2\neq 0$. Deduce that $S^{n}$ cannot be an $H$-space if $n$ is even. +\end{enumerate} +\end{enumerate} + +\chapter{Derivations of properties from the axioms} + +Returning to the axiomatic approach to cohomology, we assume given a theory on pairs +of spaces and give some deductions from the axioms. This may be viewed as a dualized +review of what we did in homology, and we generally omit the proofs. The only significant +difference that we will encounter is in the computation of the cohomology of colimits. +In a final section, we show the uniqueness of (ordinary) cohomology with coefficients +in $\pi$. + +Prior to that section, we make no use of the dimension axiom in this chapter. +A ``generalized cohomology theory'' +\index{cohomology theory!generalized} $E^*$ is defined to be a +system of functors $E^q(X,A)$ +and natural transformations $\de:E^q(A)\rtarr E^{q+1}(X,A)$ that satisfy all of our axioms +except for the dimension axiom. Similarly, we have the notion of a generalized cohomology +theory on CW pairs, and the following result holds. + +\begin{thm} A cohomology theory $E^*$ on pairs of spaces determines and is determined +by its restriction to a cohomology theory $E^*$ on pairs of CW complexes. +\end{thm} + +\section{Reduced cohomology groups and their properties} + +For a based space $X$, we define the reduced cohomology\index{reduced cohomology} of $X$ to be +$$\tilde{E}^q(X)=E^q(X,*).$$ +There results a direct sum decomposition +$$ E^*(X) \iso \tilde{E}^*(X)\oplus E^*(*)$$ +that is natural with respect to based maps. For $*\in A\subset X$, the summand $E^*(*)$ +maps isomorphically under the map $E^*(X)\rtarr E^*(A)$, and the exactness axiom implies +that there is a reduced long exact sequence +$$\cdots\rtarr \tilde{E}^{q-1}(A)\overto{\de} E^q(X,A) \rtarr \tilde{E}^q(X)\rtarr \tilde{E}^q(A) +\rtarr \cdots.$$ + +The unreduced cohomology groups are recovered as the special cases +$$ E^*(X)=\tilde{E}^*(X_+)$$ +of reduced ones, and similarly for maps. Relative cohomology groups are also special +cases of reduced ones. + +\begin{thm} +For any cofibration\index{cofibration} $i: A\rtarr X$, the quotient map $q: (X,A)\rtarr (X/A,*)$ +induces an isomorphism +$$\tilde{E}^*(X/A)=E^*(X/A,*)\iso E^*(X,A).$$ +\end{thm} + +We may replace any inclusion $i: A\rtarr X$ by the canonical cofibration $A\rtarr Mi$ +and then apply the result just given to obtain an isomorphism +$$ E^*(X,A)\iso \tilde{E}^*(Ci).$$ + +\begin{thm} For a nondegenerately based space $X$, there is a natural isomorphism +$$\SI: \tilde{E}^q(X)\iso \tilde{E}^{q+1}(\SI X).$$ +\end{thm} + +\begin{cor} Let $*\in A\subset X$, where $i: A\rtarr X$ is a cofibration between +nondegenerately based spaces. In the long exact sequence +$$\cdots\rtarr \tilde{E}^{q-1}(A)\overto{\de} \tilde{E}^q(X/A)\rtarr \tilde{E}^q(X)\rtarr +\tilde{E}^q(A)\rtarr \cdots $$ +of the pair $(X,A)$, the connecting homomorphism $\de$ is the composite +$$\tilde{E}^{q-1}(A)\overto{\SI} \tilde{E}^{q}(\SI A) \overto{\pa^*}\tilde{E}^q(X/A).$$ +\end{cor} + +\begin{cor} +For any $n$ and $q$, +$$\tilde{E}^q(S^n)\iso \tilde{E}^{q-n}(*).$$ +\end{cor} + +\section{Axioms for reduced cohomology} + +\begin{defn} A reduced cohomology theory\index{cohomology theory!reduced} $\tilde{E}^*$ consists of functors +$\tilde{E}^q$ from the homotopy category of nondegenerately based spaces +to the category of Abelian groups that satisfy the following axioms. +\begin{itemize} +\item EXACTNESS\index{exactness axiom}\ \ If $i: A\rtarr X$ is a cofibration, then the sequence +$$\tilde{E}^q(X/A)\rtarr \tilde{E}^q(X)\rtarr +\tilde{E}^q(A)$$ +is exact. +\item SUSPENSION\index{suspension axiom}\ \ +For each integer $q$, there is a natural isomorphism +$$\SI: \tilde{E}^q(X)\iso \tilde{E}^{q+1}(\SI X).$$ +\item ADDITIVITY\index{additivity axiom}\ \ +If $X$ is the wedge of a set of nondegenerately based spaces $X_i$, then +the inclusions $X_i\rtarr X$ induce an isomorphism +$$\tilde{E}^*(X) \rtarr \textstyle{\prod}_i\, \tilde{E}^*(X_i).$$ +\item WEAK EQUIVALENCE\index{weak equivalence axiom}\ \ If $f:X\rtarr Y$ is a +weak equivalence, then +$$f^*: \tilde{E}^*(Y)\rtarr \tilde{E}^*(X)$$ +is an isomorphism. +\end{itemize} +\end{defn} + +The reduced form of the dimension axiom would read +$$\tilde{H}^0(S^0)=\pi \ \ \tand \ \ \tilde{H}^q(S^0)=0\ \text{for}\ q\neq 0.$$ + +\begin{thm} A cohomology theory $E^*$ on pairs of spaces determines and is +determined by a reduced cohomology theory $\tilde{E}^*$ on nondegenerately +based spaces. +\end{thm} + +\begin{defn} A reduced cohomology theory\index{cohomology theory!reduced} $\tilde{E}^*$ on +based CW complexes consists +of functors $\tilde{E}^q$ from the homotopy category of based CW complexes +to the category of Abelian groups that satisfy the following axioms. +\begin{itemize} +\item EXACTNESS\index{exactness axiom}\ \ If $A$ is a subcomplex of $X$, then the sequence +$$\tilde{E}^q(X/A)\rtarr \tilde{E}^q(X)\rtarr +\tilde{E}^q(A)$$ +is exact. +\item SUSPENSION\index{suspension axiom}\ \ +For each integer $q$, there is a natural isomorphism +$$\SI: \tilde{E}^q(X)\iso \tilde{E}^{q+1}(\SI X).$$ +\item ADDITIVITY\index{additivity axiom}\ \ +If $X$ is the wedge of a set of based CW complexes $X_i$, then +the inclusions $X_i\rtarr X$ induce an isomorphism +$$\tilde{E}^*(X) \rtarr \textstyle{\prod}_i\, \tilde{E}^*(X_i).$$ +\end{itemize} +\end{defn} + +\begin{thm} A reduced cohomology theory $\tilde{E}^*$ on nondegenerately based spaces +determines and is determined by its restriction to a reduced cohomology theory on +based CW complexes. +\end{thm} + +\begin{thm} +A cohomology theory $E^*$ on CW pairs determines and is determined by a reduced +cohomology theory $\tilde{E}^*$ on based CW complexes. +\end{thm} + +\section{Mayer-Vietoris sequences in cohomology} + +We have Mayer-Vietoris sequences in cohomology just like those in homology. The proofs are +the same. Poincar\'{e} duality between the homology and cohomology of manifolds will be proved +by an inductive comparison of homology and cohomology Mayer-Vietoris sequences. We record two +preliminaries. + +\begin{prop} For a triple $(X,A,B)$, the following sequence is exact:\index{triple!exact sequence +of} +$$\cdots E^{q-1}(A,B) \overto{\de} E^q(X,A)\overto{j^*} E^q(X,B)\overto{i^*} +E^{q}(A,B)\rtarr \cdots.$$ +Here $i:(A,B)\rtarr (X,B)$ and $j:(X,B)\rtarr (X,A)$ are inclusions and $\de$ is the composite +$$E^{q-1}(A,B)\rtarr E^{q-1}(A)\overto{\de} E^q(X,A).$$ +\end{prop} + +Now let $(X;A,B)$ be an excisive triad and set $C=A\cap B$. + +\begin{lem} +The map +$$E^*(X,C) \rtarr E^*(A,C)\oplus E^*(B,C) $$ +induced by the inclusions of $(A,C)$ and $(B,C)$ in $(X,C)$ is an isomorphism. +\end{lem} + +\begin{thm}[Mayer-Vietoris sequence]\index{Mayer-Vietoris sequence} Let $(X;A,B)$ be an excisive +triad and set $C=A\cap B$. +The following sequence is exact: +$$\cdots \rtarr E^{q-1}(C)\overto{\DE^*} E^q(X) \overto{\ph^*} E^q(A)\oplus E^q(B)\overto{\ps^*} +E^{q}(C)\rtarr \cdots.$$ +Here, if $i: C\rtarr A$, $j: C\rtarr B$, $k: A\rtarr X$, and $\ell: B\rtarr X$ +are the inclusions, then +$$\ph^*(\ch)= (k^*(\ch),\ell^*(\ch))\ \ \tand\ \ \psi^*(\al,\be)=i^*(\al)-j^*(\be) $$ +and $\DE^*$ is the composite +$$E^{q-1}(C)\overto{\de} E^q(A,C)\iso E^q(X,B)\rtarr E^q(X).$$ +\end{thm} + +For the relative version, let $X$ be contained in some ambient space $Y$. + +\begin{thm}[Relative Mayer-Vietoris sequence]\index{Mayer-Vietoris sequence!relative} The +following sequence is exact: +$$\cdots\rtarr E^{q-1}(Y,C)\overto{\DE^*} E^q(Y,X)\overto{\ph^*} E^q(Y,A)\oplus E^q(Y,B) +\overto{\ps^*} E^{q}(Y,C)\rtarr \cdots.$$ +Here, if $i: (Y,C)\rtarr (Y,A)$, $j: (Y,C)\rtarr (Y,B)$, $k: (Y,A)\rtarr (Y,X)$, and +$\ell: (Y,B)\rtarr (Y,X)$ are the inclusions, then +$$ \ph^*(\ch)= (k^*(\ch),\ell^*(\ch)) \ \ \tand\ \ \psi^*(\al,\be)=i^*(\al)-j^*(\be)$$ +and $\DE^*$ is the composite +$$E^{q-1}(Y,C)\rtarr E^{q-1}(A,C)\iso E^{q-1}(X,B)\overto{\de} E^{q}(Y,X).$$ +\end{thm} + +\begin{cor} +The absolute and relative Mayer-Vietoris sequences are related by the following +commutative diagram: +$$\diagram +E^{q-1}(C)\rto^{\DE^*} \dto_{\de} & E^{q}(X) \rto^(0.4){\ph^*} \dto^{\de} & + E^{q}(A)\oplus E^{q}(B)\rto^(0.6){\ps^*} \dto^{\de+\de} & E^{q}(C) \dto^{\de} \\ +E^q(Y,C)\rto_(0.46){\DE^*} & E^{q+1}(Y,X) \rto_(0.35){\ph^*} & +E^{q+1}(Y,A)\oplus E^{q+1}(Y,B) \rto_(0.65){\ps^*} & E^{q+1}(Y,C).\\ +\enddiagram$$ +\end{cor} + +\section{Lim$^1$ and the cohomology of colimits} + +In this section, we let $X$ be the union of an expanding sequence of subspaces $X_i$, +$i\geq 0$. We shall use the additivity and weak equivalence axioms and the +Mayer-Vietoris sequence to explain how to compute $E^*(X)$. The answer is more subtle +than in homology because, algebraically, limits are less well behaved than colimits: +they are not exact functors from diagrams of Abelian groups to Abelian groups. Rather +than go into the general theory, we simply display how the ``first right derived functor'' +$\lim^{1}$\index{lima@$\lim^{1}$} of an inverse sequence of Abelian groups can be computed. + +\begin{lem} +Let $f_i: A_{i+1}\rtarr A_{i}$, $i\geq 1$, be a sequence of homomorphisms of Abelian groups. +Then there is an exact sequence +$$ 0\rtarr \lim\,A_i\overto{\be} \textstyle{\prod}_i A_i\overto{\al} +\textstyle{\prod}_i A_i\rtarr \lim^{1}A_i\rtarr 0,$$ +where $\al$ is the difference of the identity map and the map with coordinates $f_i$ and $\be$ +is the map whose projection to $A_i$ is the canonical map given by the definition of a limit. +\end{lem} + +That is, we may as well define $\lim^{1}A_i$ to be the displayed cokernel. We then have the +following result. + +\begin{thm}\index{colimit!cohomology of} +For each $q$, there is a natural short exact sequence +$$0 \rtarr {\lim}^{1}\,E^{q-1}(X_i)\rtarr E^q(X)\overto{\pi} \lim\,E^q(X_i) \rtarr 0,$$ +where $\pi$ is induced by the inclusions $X_i\rtarr X$. +\end{thm} +\begin{proof} +We use the notations and constructions in the proof that homology commutes with +colimits and consider the excisive triad $(\tel\,X_i;A,B)$ with $C=A\cap B$ constructed +there. By the additivity axiom, +$$E^*(A)=\textstyle{\prod}_i\, E^*(X_{2i}),\ \ E^*(B) +=\textstyle{\prod}_i\, E^*(X_{2i+1}),\ \tand\ E^*(C)=\textstyle{\prod}_i\, E^*(X_i).$$ +We construct the following commutative diagram, whose top row is the cohomology Mayer-Vietoris +sequence of the triad $(\tel\, X_i;A,B)$ and whose bottom row is an exact sequence +of the sort displayed in the previous lemma. +\begin{footnotesize} +$$\diagram +\cdots \rto & E^q(\tel X_i) \rto \dto_{\iso} & E^q(A)\oplus E^q(B) \rto \dto_{\iso} + & E^q(C) \rto \dto^{\iso} & E^{q+1}(\tel\,X_i) \dto^{\iso} \rto & \cdots \\ +\cdots \rto & E^q(X) \rto^{\be'} \dto_{\pi'} +& \prod E^q(X_i) \rto^{\al'} \dto_{\prod(-1)^{i}} + & \prod_i E^q(X_i) \dto^{\prod_i(-1)^{i}} \rto & E^{q+1}(X) \rto & \cdots \\ +0 \rto & \lim\,E^q(X_i) \rto^{\be} & \prod_i E^q(X_i) \rto^{\al} + & \prod_i E^q(X_i) \rto & \lim^1 E^q(X_i) \rto & 0. \\ +\enddiagram$$ +\end{footnotesize} +The commutativity of the bottom middle square is a comparison based on the sign used in +the Mayer-Vietoris sequence. Here the map $\pi'$ differs by alternating signs from the +canonical map $\pi$, but this does not affect the conclusion. A chase of the diagram +implies the result. +\end{proof} + +The $\lim^1$ ``error terms'' are a nuisance, and it is important to know when they vanish. +We say that an inverse sequence $f_i: A_{i+1}\rtarr A_i$ satisfies the Mittag-Leffler +condition\index{Mittag-Leffler condition} if, for each fixed $i$, there exists $j\geq i$ such +that, for every $k>j$, the +image of the composite $A_k\rtarr A_i$ is equal to the image of the composite $A_j\rtarr A_i$. +For example, this holds if all but finitely many of the $f_i$ are epimorphisms or if +the $A_i$ are all finite. As a matter of algebra, we have the following vanishing result. + +\begin{lem} +If the inverse sequence $f_i: A_{i+1}\rtarr A_i$ satisfies the Mittag-Leffler condition, +then $\lim^1\,A_i=0$. +\end{lem} + +For example, for $qn$, and $\tilde{H}_n(M;\pi)=0$ +if $M$ is connected and is not compact. +\end{thm} + +We can use this together with Mayer-Vietoris sequences to construct $R$-fun\-da\-men\-tal classes +at compact subspaces from $R$-orientations. To avoid trivialities, we tacitly assume that $n>0$. +(The trivial case $n=0$ forced the use of reduced homology in the statement; where arguments use +reduced homology below, it is only to ensure that what we write is correct in dimension zero.) + +\begin{thm} +Let $K$ be a compact subset of $M$. Then, for any coefficient group $\pi$, $H_i(M,M-K;\pi)=0$ +if $i>n$, and an $R$-orientation of $M$ determines an $R$-fundamental class of $M$ at $K$. +In particular, if $M$ is compact, then an $R$-orientation of $M$ determines an $R$-fundamental +class of $M$. +\end{thm} +\begin{proof} +First assume that $K$ is contained in a coordinate chart $U\iso \bR^n$. By excision and +exactness, we then have +$$H_i(M,M-K;\pi)\iso H_i(U,U-K;\pi)\iso \tilde{H}_{i-1}(U-K;\pi).$$ +Since $U-K$ is open in $U$, the vanishing theorem implies that $\tilde{H}_{i-1}(U-K;\pi)=0$ +for $i>n$. In fact, a lemma used in the proof of the vanishing theorem will prove this +directly. In this case, an $R$-fundamental class in $H_n(M,M-U)$ maps to an $R$-fundamental +class in $H_n(M,M-K)$. A general compact subset $K$ of $M$ can be written as the union of +finitely many compact subsets, each of which is contained in a coordinate chart. By induction, +it suffices to prove the result for $K\cup L$ under the assumption that it holds for $K$, $L$, +and $K\cap L$. With any coefficients, we have the Mayer-Vietoris sequence +\begin{multline*} +\cdots \rtarr H_{i+1}(M,M-K\cap L)\overto{\DE}H_i(M,M-K\cup L)\\ +\overto{\ps}H_i(M,M-K)\oplus H_i(M,M-L)\overto{\ph} H_i(M,M-K\cap L) \rtarr \cdots. +\end{multline*} +The vanishing of $H_i(M,M-K\cup L;\pi)$ for $i>n$ follows directly. Now take $i=n$ and +take coefficients in $R$. Then $\ps$ is a monomorphism. The $R$-fun\-da\-men\-tal classes +$z_K\in H_n(M,M-K)$ and $z_L\in H_n(M,M-L)$ determined by a given $R$-orientation both map +to the $R$-fundamental class $z_{K\cap L}\in H_n(M,M-K\cap L)$ determined by the given +$R$-orientation. Therefore +$$\ph(z_K,z_L) = z_{K\cap L}- z_{K\cap L} = 0$$ +and there exists a unique $z_{K\cup L}\in H_n(M,M-K\cup L)$ such that +$$ \ps(z_{K\cup L}) = (z_K,z_L).$$ +Clearly $z_{K\cup L}$ is an $R$-fundamental class of $M$ at $K\cup L$. +\end{proof} + +The vanishing theorem also implies the following dichotomy, which we have already noticed +in our examples of explicit calculations. + +\begin{cor} +Let $M$ be a connected compact $n$-manifold, $n>0$. Then either $M$ is not orientable and +$H_n(M;\bZ)=0$ or $M$ is orientable\index{orientable} and the map +$$H_n(M;\bZ) \rtarr H_n(M,M-x;\bZ)\iso \bZ$$ +is an isomorphism for every $x\in M$. +\end{cor} +\begin{proof} +Since $M-x$ is connected and not compact, $H_n(M-x;\pi)=0$ and thus +$$H_n(M;\pi) \rtarr H_n(M,M-x;\pi)\iso \pi$$ +is a monomorphism for all coefficient groups $\pi$. In particular, by the universal +coefficient theorem, +$$H_n(M;\bZ)\ten \bZ_q \rtarr H_n(M,M-x;\bZ)\ten \bZ_q\iso \bZ_q$$ +is a monomorphism for all positive integers $q$. If $H_n(M;\bZ)\neq 0$, then +$H_n(M;\bZ)\iso \bZ$ with generator mapped to some multiple of a generator of +$H_n(M,M-x;\bZ)$. By the mod $q$ monomorphism, the coefficient must be $\pm 1$. +\end{proof} + +As an aside, the corollary leads to a striking example of the failure of the naturality +of the splitting in the universal coefficient theorem. Consider a connected, compact, +non-orientable $n$-manifold $M$. Let $x\in M$ and write $M_x$ for the pair $(M,M-x)$. +Since $M$ is $\bZ_2$-orientable, the middle vertical arrow in the following diagram is +an isomorphism between copies of $\bZ_2$: +$$\diagram +0 \rto & H_n(M)\ten \bZ_2 \rto \dto_0 & H_n(M;\bZ_2) \rto \dto^{\iso} +& \Tor_1^{\bZ}(H_{n-1}(M),\bZ_2) \rto \dto^0 & 0 \\ +0 \rto & H_n(M_x)\ten \bZ_2 \rto & H_n(M_x;\bZ_2) \rto +& \Tor_1^{\bZ}(H_{n-1}(M_x),\bZ_2) \rto & 0. \\ +\enddiagram$$ +Clearly $H_{n-1}(M,M-x)=0$, and the corollary gives that $H_n(M)=0$. Thus the left and +right vertical arrows are zero. If the splittings of the rows were natural, this would +imply that the middle vertical arrow is also zero. + +\section{The proof of the vanishing theorem} + +Let $M$ be an $n$-manifold, $n>0$. Take all homology groups with coefficients in a given +Abelian group $\pi$ in this section. We must prove the intuitively obvious statement that +$H_i(M)=0$ for $i>n$ and the much more subtle statement that $H_n(M)=0$ if $M$ is +connected and is not compact. The last statement is perhaps the technical heart of our +proof of the Poincar\'e duality theorem. + +We begin with the general observation that homology is ``compactly +supported''\index{compactly supported homology} in the +sense of the following result. + +\begin{lem} For any space $X$ and element $x\in H_q(X)$, there is a compact subspace +$K$ of $X$ and an element $k\in H_q(K)$ that maps to $x$. +\end{lem} +\begin{proof} +Let $\ga: Y\rtarr X$ be a CW approximation of $X$ and let $x=\ga_*(y)$. If $y$ is represented +by a cycle $z\in C_q(Y)$, then $z$, as a finite linear combination of $q$-cells, is an +element of $C_q(L)$ for some finite subcomplex $L$ of $Y$. Let $K=\ga(L)$ and let $k$ be +the image of the homology class represented by $z$. Then $K$ is compact and $k$ maps to $x$. +\end{proof} + +We need two lemmas about open subsets of $\bR^n$ to prove the vanishing theorem, the first +of which is just a special case. + +\begin{lem} If $U$ is open in $\bR^n$, then $H_i(U)=0$ for $i\geq n$. +\end{lem} +\begin{proof} +Let $s\in H_i(U)$, $i\geq n$. There is a compact subspace $K$ of $U$ and an element +$k\in H_i(K)$ that maps to $s$. We may decompose $\bR^n$ as a CW complex whose $n$-cells +are small $n$-cubes in such a way that there is a finite subcomplex $L$ of $\bR^n$ with +$K\subset L\subset U$. (To be precise, use a cubical grid with small enough mesh.) For $i>0$, +the connecting homomorphisms $\pa$ are isomorphisms in the commutative diagram +$$\diagram +H_{i+1}(\bR^n,L) \rto \dto_{\pa} & H_{i+1}(\bR^n,U) \dto^{\pa} \\ +H_i(L) \rto & H_i(U).\\ +\enddiagram$$ +Since $(\bR^n,L)$ has no relative $q$-cells for $q > n$, the groups on the left are zero for +$i\geq n$. Since $s$ is in the image of $H_i(L)$, $s=0$. +\end{proof} +\begin{lem} +Let $U$ be open in $\bR^n$. Suppose that $t\in H_n(\bR^n,U)$ maps to zero in $H_n(\bR^n,\bR^n-x)$ +for all $x\in \bR^n-U$. Then $t=0$. +\end{lem} +\begin{proof} +We prove the equivalent statement that if $s\in \tilde{H}_{n-1}(U)$ maps to zero in +$\tilde{H}_{n-1}(\bR^n-x)$ for all $x\in \bR^n-U$, then $s=0$. Choose a compact subspace +$K$ of $U$ such that $s$ is in the image of $\tilde{H}_{n-1}(K)$. Then $K$ is contained +in an open subset $V$ whose closure $\bar{V}$ is compact and contained in $U$, hence $s$ +is the image of an element $r\in\tilde{H}_{n-1}(V)$. We claim that $r$ maps to zero in +$\tilde{H}_{n-1}(U)$, so that $s=0$. Of course, $r$ maps to zero in +$\tilde{H}_{n-1}(\bR^n-x)$ if $x\not\!\!{\in}\,U$. Let $T$ be an open contractible subset of $\bR^n$ +such that $\bar{V}\subset T$ and $\bar{T}$ is compact. For example, $T$ could be a large +enough open cube. Let $L=T-(T\cap U)$. For each $x\in \bar{L}$, choose a closed cube $D$ that +contains $x$ and is disjoint from $V$. A finite set $\sset{D_1,\ldots\!,D_q}$ of these +cubes covers $\bar{L}$. Let $C_i=D_i\cap T$ and observe that $(\bR^n-D_i)\cap T = T-C_i$. +We see by induction on $p$ that $r$ maps to zero in $\tilde{H}_{n-1}(T-(C_1\cup\cdots\cup C_p))$ +for $0\leq p\leq q$. This is clear if $p=0$. For the inductive step, observe that +$$T-(C_1\cup\cdots\cup C_p) = (T-(C_1\cup\cdots\cup C_{p-1}))\cap (\bR^n-D_p)$$ +and that $H_n((T-(C_1\cup\cdots\cup C_{p-1}))\cup (\bR^n-D_p)) = 0$ by the previous lemma. +Therefore the map +$$\tilde{H}_{n-1}(T-(C_1\cup\cdots\cup C_p))\rtarr +\tilde{H}_{n-1}(T-(C_1\cup\cdots\cup C_{p-1}))\oplus \tilde{H}_{n-1}(\bR^n-D_p)$$ +in the Mayer-Vietoris sequence is a monomorphism. Since $r\in\tilde{H}_{n-1}(V)$ maps to +zero in the two right-hand terms, by the induction hypothesis and the contractibility of +$D_p$ to a point $x\not\!\!{\in}\,U$, it maps to zero in the left-hand term. Since +$$V\subset T-(C_1\cup\cdots\cup C_q)\subset T\cap U\subset U,$$ +this implies our claim that $r$ maps to zero in $\tilde{H}_{n-1}(U)$. +\end{proof} + +\begin{proof}[Proof of the vanishing theorem] +Let $s\in H_i(M)$. We must prove that $s=0$ if $i>n$ and if $i=n$ when $M$ is connected +and not compact. Choose a compact subspace $K$ of $M$ such that $s$ is in the image of +$H_i(K)$. Then $K$ is contained in some finite union $U_1\cup\cdots\cup U_q$ of +coordinate charts, and it suffices to prove that $H_i(U_1\cup\cdots\cup U_q)=0$ for the +specified values of $i$. Inductively, using that $H_i(U)=0$ for $i\geq n$ when $U$ is +an open subset of a coordinate chart, it suffices to prove that $H_i(U\cup V)=0$ for +the specified values of $i$ when $U$ is a coordinate chart and $V$ is an open subspace +of $M$ such that $H_i(V)=0$ for the specified values of $i$. We have the Mayer-Vietoris +sequence +$$H_i(U)\oplus H_i(V) \rtarr H_i(U\cup V) \rtarr \tilde{H}_{i-1}(U\cap V) \rtarr +\tilde{H}_{i-1}(U)\oplus \tilde{H}_{i-1}(V).$$ +If $i>n$, the vanishing of $H_i(U\cup V)$ follows immediately. Thus assume that $M$ is +connected and not compact and consider the case $i=n$. We have $H_n(U)=0$, $H_n(V)=0$, +and $\tilde{H}_{n-1}(U)=0$. It follows that $H_n(U\cup V)=0$ if and only if +$i_*: \tilde{H}_{n-1}(U\cap V)\rtarr \tilde{H}_{n-1}(V)$ is a monomorphism, where +$i: U\cap V\rtarr V$ is the inclusion. + +We claim first that $H_n(M)\rtarr H_n(M,M-y)$ is the zero homomorphism for any $y\in M$. +If $x\in M$ and $L$ is a path in $M$ connecting $x$ to $y$, then the diagram +$$\diagram +& & H_n(M,M-x) \\ +H_n(M) \rto & H_n(M,M-L) \urto^{\iso} \drto_{\iso} & \\ +& & H_n(M,M-y) \\ +\enddiagram$$ +shows that if $s\in H_n(M)$ maps to zero in $H_n(M,M-x)$, then it maps to zero in $H_n(M,M-y)$. +If $s$ is in the image of $H_n(K)$ where $K$ is compact, we may choose a point $x\in M-K$. +Then the map $K\rtarr M\rtarr (M,M-x)$ factors through $(M-x,M-x)$ and therefore $s$ maps to +zero in $H_n(M,M-x)$. This proves our claim. + +Now consider the following diagram, where $y\in U- U\cap V$: +$$\diagram +& & H_n(U\cup V) \rto \dlto & H_n(M) \dto^{0}\\ +H_n(V,U\cap V)\dto_{\pa} \rto & H_n(U\cup V, U\cap V) \dlto^{\pa} \rrto & & H_n(M,M-y) \\ +\tilde{H}_{n-1}(U\cap V) \dto_{i_*} & H_n(U,U\cap V) \lto^{\pa} \rrto \uto +& & H_n(U,U-y) \uto_{\iso}\\ +\tilde{H}_{n-1}(V). & & & \\ +\enddiagram$$ +Let $r\in \ker\,i_*$. Since $\tilde{H}_{n-1}(U)=0$, the bottom map $\pa$ is an epimorphism +and there exists $s\in H_n(U,U\cap V)$ such that $\pa(s)=r$. We claim that $s$ maps to zero +in $H_n(U,U-y)$ for every $y\in U-(U\cap V)$. By the previous lemma, this will imply that +$s=0$ and thus $r=0$, so that $i_*$ is indeed a monomorphism. Since $i_*(r)=0$, there exists +$t\in H_n(V,U\cap V)$ such that $\pa(t)=r$. Let $s'$ and $t'$ be the images of $s$ +and $t$ in $H_n(U\cup V,U\cap V)$. Then $\pa(s'-t')=0$, hence there exists $w\in H_n(U\cup V)$ +that maps to $s'-t'$. Since $w$ maps to zero in $H_n(M,M-y)$, so does $s'-t'$. Since the map +$(V,U\cap V)\rtarr (M,M-y)$ factors through $(M-y,M-y)$, $t$ and thus also $t'$ maps to zero +in $H_n(M,M-y)$. Therefore $s'$ maps to zero in $H_n(M,M-y)$ and thus $s$ maps to zero in +$H_n(U,U-y)$, as claimed. +\end{proof} + +\section{The proof of the Poincar\'e duality theorem} + +Let $M$ be an $R$-oriented $n$-manifold, not necessarily compact. Unless otherwise specified, +we take homology and cohomology with coefficients in a given $R$-module $\pi$ in this section. +Remember that homology +is a covariant functor with compact supports. Cohomology is a contravariant functor, and +it does not have compact supports. We would like to prove the Poincar\'e duality theorem +by inductive comparisons of Mayer-Vietoris sequences, and the opposite variance of +homology and cohomology makes it unclear how to proceed. To get around this, we introduce +a variant of cohomology that does have compact supports and has enough covariant functoriality +to allow us to proceed by comparisons of Mayer-Vietoris sequences. + +Consider the set $\sK$ of compact subspaces $K$ of $M$. This set is directed under inclusion; to +conform with our earlier discussion of colimits, we may view $\sK$ as a category whose objects +are the compact subspaces $K$ and whose maps are the inclusions between them. We define +$$H^q_c(M) = \colim H^q(M,M-K),$$ +where the colimit is taken with respect to the homomorphisms +$$H^q(M,M-K) \rtarr H^q(M,M-L)$$ +induced by the inclusions $(M,M-L)\subset (M,M-K)$ for $K\subset L$. This is the cohomology of +$M$ with compact supports.\index{cohomology with compact +supports}\index{compactly supported cohomology} Intuitively, thinking in +terms of singular cohomology, its elements +are represented by cocycles that vanish off some compact subspace. + +A map $f:M\rtarr N$ is said to be proper if $f^{-1}(L)$ is compact in $M$ when $L$ is compact +in $N$. This holds, for example, if $f$ is the inclusion of a closed subspace. For such $f$, +we obtain an induced homomorphism $f^*: H^*_c(N)\rtarr H^*_c(M)$ in an evident way. However, +we shall make no use of this contravariant functoriality. + +What we shall use is a kind of covariant functoriality that will allow us to compare long +exact sequences in homology and cohomology. Explicitly, for an open subspace $U$ of $M$, we +obtain a homomorphism $H^q_c(U)\rtarr H^q_c(M)$ by passage to colimits from the excision +isomorphisms +$$ H^q(U,U-K) \rtarr H^q(M,M-K)$$ +for compact subspaces $K$ of $U$. + +For each compact subspace $K$ of $M$, the $R$-orientation of $M$ determines a fundamental +class $z_K\in H_n(M,M-K;R)$. Taking the relative cap product with $z_K$, we obtain a duality +homomorphism +$$D_K: H^p(M,M-K) \rtarr H_{n-p}(M).$$ +If $K\subset L$, the following diagram commutes: +$$\diagram +H^p(M,M-K) \rrto \drto_{D_K} & & H^p(M,M-L) \dlto^{D_L}\\ +& H_{n-p}(M). & \\ +\enddiagram$$ +We may therefore pass to colimits to obtain a duality homomorphism +$$D: H^p_c(M) \rtarr H_{n-p}(M).$$ +If $U$ is open in $M$ and is given the induced $R$-orientation, then the following naturality +diagram commutes: +$$\diagram + H^p_c(U) \dto \rto^(0.43){D} & H_{n-p}(U) \dto \\ + H^p_c(M) \rto_(0.4){D} & H_{n-p}(M).\\ +\enddiagram$$ +If $M$ itself is compact, then $M$ is cofinal among the compact subspaces of $M$. Therefore +$H^p_c(M) = H^p(M)$, and the present duality map $D$ coincides with that of the Poincar\'e +duality theorem as originally stated. We shall prove a generalization to not necessarily +compact manifolds. + +\begin{thm}[Poincar\'e duality]\index{Poincare duality theorem@Poincar\'e duality theorem} +Let $M$ be an $R$-oriented $n$-manifold. Then \linebreak +$D: H^p_c(M)\rtarr H_{n-p}(M)$ is an isomorphism. +\end{thm} +\begin{proof} +We shall prove that $D: H^p_c(U)\rtarr H_{n-p}(U)$ is an isomorphism for every open +subspace $U$ of $M$. The proof proceeds in five steps. +\begin{proof}[Step 1] {\em The result holds for any coordinate chart $U$.}\\ +We may take $U=M=\bR^n$. The compact cubes $K$ are cofinal among the compact subspaces of $\bR^n$. +For such $K$ and for $x\in K$, +$$H^p(\bR^n,\bR^n-K)\iso H^p(\bR^n,\bR^n-x)\iso \tilde{H}^{p-1}(S^{n-1})\iso \tilde{H}^p(S^n).$$ +The maps of the colimit system defining $H^p_c(\bR^n)$ are clearly isomorphisms. By the definition +of the cap product, we see that $D: H^n(\bR^n,\bR^n-x)\rtarr H_0(\bR^n)$ is an isomorphism. +Therefore $D_K$ is an isomorphism for every compact cube $K$ and so +$D: H^n_c(\bR^n)\rtarr H_0(\bR^n)$ is an isomorphism. +\end{proof} +\begin{proof}[Step 2] {\em If the result holds for open subspaces $U$ and $V$ and their +intersection, then it holds for their union.}\\ +Let $W=U\cap V$ and $Z=U\cup V$. The compact subspaces of $Z$ that are unions of a compact +subspace $K$ of $U$ and a compact subspace $L$ of $V$ are cofinal among all of the compact +subspaces of $Z$. For such $K$ and $L$, we have the following commutative diagram with +exact rows. We let $J=K\cap L$ and $N=K\cup L$, and we write $U_K = (U,U-K)$, and so on, +to abbreviate notation. +\begin{small} +$$\diagram +\rto & H^p(Z_J) \rto \dto_{\iso}& H^p(Z_K)\oplus H^p(Z_L) \rto \dto^{\iso} +& H^p(Z_N) \rto \ddouble & H^{p+1}(Z_J) \rto \dto^{\iso} & \\ +\rto & H^p(W_J) \rto \dto_D & H^p(U_K)\oplus H^p(V_L) \rto \dto^{D\oplus D} +& H^p(Z_N) \rto \dto^D & H^{p+1}(W_J) \rto \dto^D & \\ +\rto & H_{n-p}(W) \rto & H_{n-p}(U)\oplus H_{n-p}(V) \rto & H_{n-p}(Z) \rto & H_{n-p-1}(W)\rto &\\ +\enddiagram$$ +\end{small} +The top row is the relative Mayer-Vietoris sequence of the triad $(Z;Z-K,Z-L)$. +The middle row results from the top row by excision isomorphisms. The bottom row is the +absolute Mayer-Vietoris sequence of the triad $(Z;U,V)$. The left two squares commute by +naturality. The right square commutes by a diagram chase from the definition of the +cap product. The entire diagram is natural with respect to pairs $(K,L)$. We obtain a +commutative diagram with exact rows on passage to colimits, and the conclusion follows +by the five lemma. +\end{proof} +\begin{proof}[Step 3] {\em If the result holds for each $U_i$ in a totally ordered set of +open subspaces $\sset{U_i}$, then it holds for the union $U$ of the $U_i$.}\\ +Any compact subspace $K$ of $U$ is contained in a finite union of the $U_i$ and therefore +in one of the $U_i$. Since homology is compactly supported, it follows that +$\colim H_{n-p}(U_i)\iso H_{n-p}(U)$. On the cohomology side, we have +\begin{eqnarray*} +\colim_i\,H^p_c(U_i) & = & \colim_i\colim_{\sset{K| K\subset U_i}} H^p(U_i,U_i-K) \\ +& \iso & \colim_{\sset{K\subset U}}\colim_{\sset{i|K\subset U_i}} H^p(U_i,U_i-K) \\ +& \iso & \colim_{\sset{K\subset U}} H^p(U,U-K) = H^p_c(U). +\end{eqnarray*} +Here the first isomorphism is an (algebraic) interchange of colimits isomorphism: both +composite colimits are isomorphic to $\colim H^p_c(U_i,U_i-K)$, where the colimit +runs over the pairs $(K,i)$ such that $K\subset U_i$. The second isomorphism holds +since $\colim_{\sset{i|K\subset U_i}} H^p(U_i,U_i-K)\iso H^p(U,U-K)$ because the +colimit is taken over a system of inverses of excision isomorphisms. The conclusion +follows since a colimit of isomorphisms is an isomorphism. +\end{proof} +\begin{proof}[Step 4] {\em The result holds if $U$ is an open subset of a coordinate +neighborhood.}\\ +We may take $M=\bR^n$. If $U$ is a convex subset of $\bR^n$, then $U$ is homeomorphic +to $\bR^n$ and Step 1 applies. Since the intersection of two convex sets is convex, +it follows by induction from Step 2 that the conclusion holds for any finite union of +convex open subsets of $\bR^n$. Any open subset $U$ of $\bR^n$ is the union of countably +many convex open subsets. By ordering them and letting $U_i$ be the union of the first $i$, +we see that the conclusion for $U$ follows from Step 3. +\end{proof} +\begin{proof}[Step 5] {\em The result holds for any open subset $U$ of $M$}.\\ +We may as well take $M=U$. By Step 3, we may apply Zorn's lemma to conclude that there is +a maximal open subset $V$ of $M$ for which the conclusion holds. If $V$ is not all of $M$, +say $x\not\in V$, we may choose a coordinate chart $U$ such that $x\in U$. +By Steps 2 and 4, the result holds for $U\cup V$, contradicting the maximality of $V$. +\end{proof} +This completes the proof of the Poincar\'e duality theorem. \end{proof} + +\section{The orientation cover} + +There is an orientation cover\index{orientation cover} of a manifold that helps illuminate the +notion of orientability. For the moment, we relax the requirement that the total +space of a cover be connected. Here we take homology with integer coefficients. + +\begin{prop} +Let $M$ be a connected $n$-manifold. Then there is a $2$-fold cover $p:\tilde{M}\rtarr M$ +such that $\tilde{M}$ is connected if and only if $M$ is not orientable.\index{orientable} +\end{prop} +\begin{proof} +Define $\tilde{M}$ to be the set of pairs $(x,\al)$, where $x\in M$ and where +$\al\in H_n(M,M-x)\iso \bZ$ is a generator. Define $p(x,\al)=x$. If $U\subset M$ is +open and $\be\in H_n(M,M-U)$ is a fundamental class of $M$ at $U$, define +$$\langle U,\be\rangle = \sset{(x,\al)|x\in U \tand \be \ \text{maps to}\ \al}.$$ +The sets $\langle U,\be\rangle$ form a base for a topology on $\tilde{M}$. In fact, if +$(x,\al)\in \langle U,\be\rangle \cap \langle V,\ga\rangle$, we can choose a coordinate +neighborhood $W\subset U\cap V$ such that $x\in W$. There is a unique class $\al'\in H_n(M,M-W)$ +that maps to $\al$, and both $\be$ and $\ga$ map to $\al'$. Therefore +$$\langle W,\al'\rangle \subset \langle U,\be\rangle \cap \langle V,\ga\rangle.$$ +Clearly $p$ maps $\langle U,\be\rangle$ homeomorphically onto $U$ and +$$p^{-1}(U) = \langle U,\be\rangle \cup \langle U,-\be\rangle.$$ +Therefore $\tilde{M}$ is an $n$-manifold and $p$ is a $2$-fold cover. Moreover, $\tilde{M}$ +is oriented. Indeed, if $U$ is a coordinate chart and $(x,\al)\in \langle U,\be\rangle$, +then the following maps all induce isomorphisms on passage to homology: +$$\diagram +(\tilde{M},\tilde{M}-\langle U,\be\rangle) \dto & (M,M-U) \dto \\ +(\tilde{M},\tilde{M}-(x,\al)) & (M,M-x) \\ +(\langle U,\be\rangle, \langle U,\be\rangle -(x,\al)) \uto \rto^(0.6)p_(0.6){\iso} & (U,U-x). \uto \\ +\enddiagram$$ +Via the diagram, $\be\in H_n(M,M-U)$ specifies an element +$\tilde{\be}\in H_n(\tilde{M},\tilde{M}-\langle U,\be\rangle)$, and $\tilde{\be}$ is +independent of the choice of $(x,\al)$. These classes are easily seen to specify an +orientation of $\tilde{M}$. Essentially by definition, an orientation of $M$ is a +cross section $s: M \rtarr \tilde{M}$: if $s(U) = \langle U,\be\rangle$, then these +$\be$ specify an orientation. Given one section $s$, changing the signs of the $\be$ +gives a second section $-s$ such that $\tilde{M}= \im(s)\amalg \im(-s)$, showing that +$\tilde{M}$ is not connected if $M$ is oriented. +\end{proof} + +The theory of covering spaces gives the following consequence. + +\begin{cor} If $M$ is simply connected, or if $\pi_1(M)$ contains no subgroup of +index $2$, then $M$ is orientable. If $M$ is orientable, then $M$ admits exactly +two orientations. +\end{cor} +\begin{proof} +If $M$ is not orientable, then $p_*(\pi_1(\tilde{M}))$ is a subgroup of $\pi_1(M)$ +of index $2$. This implies the first statement, and the second statement is clear. +\end{proof} + +We can use homology with coefficients in a commutative ring $R$ to construct an +analogous $R$-orientation cover.\index{Rorientation cover@$R$-orientation cover} It +depends on the units of $R$. For +example, if $R=\bZ_2$, then the $R$-orientation cover is the identity map of $M$ +since there is a unique unit in $R$. This reproves the obvious fact that any +manifold is $\bZ_2$-oriented. The evident ring homomorphism $\bZ\rtarr R$ induces +a natural homomorphism $H_*(X;\bZ)\rtarr H_*(X;R)$, and we see immediately that +an orientation of $M$ induces an $R$-orientation of $M$ for any $R$. + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Prove: there is no homotopy equivalence +$f: \bC P^{2n}\rtarr \bC P^{2n}$ that reverses orientation +(induces multiplication by $-1$ on $H_{4n}(\bC P^{2n})$). +\end{enumerate} + +In the problems below, $M$ is assumed to be a compact connected +$n$-manifold (without boundary), where $n\geq 2$. +\begin{enumerate} +\item[2.] Prove that if $M$ is a Lie group, then $M$ is orientable. +\item[3.] Prove that if $M$ is orientable, then $H_{n-1}(M; \bZ)$ is a free Abelian group. +\item[4.] Prove that if $M$ is not orientable, then the torsion subgroup of $H_{n-1}(M; \bZ)$ is +cyclic of order $2$ and $H_n(M; \bZ_q)$ is zero if $q$ is odd and is cyclic +of order $2$ if $q$ is even. (Hint: use universal coefficients and the transfer +homomorphism of the orientation cover.) +\item[5.] Let $M$ be oriented with fundamental class $z$. Let $f: S^n\rtarr M$ be a +map such that $f_*(i_n) = qz$, where $i_n \in H_n(S^n; \bZ)$ is the fundamental +class and $q \neq 0$. +\begin{enumerate} +\item[(a)] Show that $f_*: H_*(S^n; \bZ_p) \rtarr H_*(M; \bZ_p)$ is an isomorphism +if $p$ is a prime that does not divide $q$. +\item[(b)] Show that multiplication by $q$ annihilates $H_i(M; \bZ)$ if +$1 \leq i \leq n-1$. +\end{enumerate} +\item[6.] +\begin{enumerate} +\item[(a)] Let $M$ be a compact $n$-manifold. Suppose that $M$ is +homotopy equivalent to $\SI Y$ for some connected based space $Y$. Deduce that $M$ +has the same integral homology groups as $S^n$. (Hint: use the vanishing of cup products +on $\tilde H^*(\SI Y)$ and Poincar\'e duality, treating the cases $M$ orientable +and $M$ non-orientable separately.) +\item[(b)] Deduce that $M$ is homotopy equivalent to $S^n$. Does it follow that $Y$ is homotopy +equivalent to $S^{n-1}$? +\end{enumerate} +\item[7.]* Essay: The singular cohomology $H^*(M;\bR)$ is isomorphic to the de Rham +cohomology of $M$. Why is this plausible? Sketch proof? +\end{enumerate} + +\clearpage + +\thispagestyle{empty} + +\chapter{The index of manifolds; manifolds with boundary} + +The Poincar\'e duality theorem imposes strong constraints on the Euler +characteristic of a manifold. It also leads to new invariants, most +notably the index. Moreover, there is a relative version of Poincar\'e +duality in the context of manifolds with boundary, and this leads to +necessary algebraic conditions on the cohomology of a manifold that must be +satisfied if it is to be a boundary. In particular, the index of a compact +oriented $4n$-manifold $M$ is zero if $M$ is a boundary. We shall later +outline the theory of cobordism, which leads to necessary {\em and sufficient} +algebraic conditions for a manifold to be a boundary. + +\section{The Euler characteristic of compact manifolds} + +The Euler characteristic\index{Euler characteristic!of a space} $\ch (X)$ of a space with +finitely generated homology is defined by +$$\ch (X) = \textstyle{\sum}_i (-1)^i \ \text{rank}\ H_i(X;\bZ).$$ +The universal coefficient theorem implies that +$$\ch (X) = \textstyle{\sum}_i (-1)^i \dim H_i(X;F)$$ +for {\em any} field of coefficients $F$. Examination of the relevant short +exact sequences shows that +$$\ch (X) = \textstyle{\sum}_i (-1)^i \ \text{rank}\ C_i(X;\bZ)$$ +for {\em any} decomposition of $X$ as a finite CW complex. The verifications +of these statements are immediate from earlier exercises. + +Now consider a compact oriented $n$-manifold. Recall that we take it for granted +that $M$ can be decomposed as a finite CW complex, so that each $H_i(M;\bZ)$ is +finitely generated. By the universal coefficient theorem and Poincar\'e duality, +we have +$$H_i(M;F)\iso H^i(M;F)\iso H_{n-i}(M;F)$$ +for any field $F$. We may take $F=\bZ_2$, and so dispense with the requirement +that $M$ be oriented. If $n$ is odd, the summands of $\ch(M)$ cancel in pairs, +and we obtain the following conclusion. + +\begin{prop} If $M$ is a compact manifold of odd dimension, then $\ch(M)=0$. +\end{prop} + +If $n=2m$ and $M$ is oriented, then +$$\ch(M) = \textstyle{\sum}_{i=0}^{m-1} (-1)^i 2 \dim H_i(M) + (-1)^m \dim H_m(M)$$ +for any field $F$ of coefficients. Let us take $F=\bQ$. Of course, we can replace +homology by cohomology in the definition and formulas for $\ch(M)$. The middle +dimensional cohomology group $H^m(M)$ plays a particularly important role. Recall +that we have the cup product pairing\index{cup product pairing} +$$\ph: H^m(M)\ten H^m(M) \rtarr \bQ$$ +specified by $\ph(\al,\be) = \langle\al\cup\be,z\rangle$. This pairing is nonsingular. +Since $\al\cup \be =(-1)^m\be\cup\al$, it is skew symmetric if $m$ is odd and is +symmetric if $m$ is even. When $m$ is odd, we obtain the following conclusion. + +\begin{prop} +If $M$ is a compact oriented $n$-manifold, where $n\equiv 2\ \text{mod}\ 4$, then +$\ch(M)$ is even. +\end{prop} +\begin{proof} It suffices to prove that $\dim H^{m}(M)$ is even, where $n=2m$, and this is +immediate from the following algebraic observation. +\end{proof} + +\begin{lem} +Let $F$ be a field of characteristic $\neq 2$, $V$ be a finite dimensional vector +space over $F$, and $\ph: V\times V \rtarr F$ be a nonsingular skew symmetric bilinear form. +Then $V$ has a basis $\sset{x_1,\ldots\!,x_r, y_1,\ldots\!,y_r}$ such that $\ph(x_i,y_i)=1$ for +$1\leq i\leq r$ and $\ph(z,w)=0$ for all other pairs of basis elements $(z,w)$. Therefore +the dimension of $V$ +is even. +\end{lem} +\begin{proof} +We proceed by induction on $\dim V$, and we may assume that $V\neq 0$. Since +$\ph(x,y)=-\ph(y,x)$, $\ph(x,x)=0$ for all +$x\in V$. Choose $x_1\neq 0$. Certainly there exists $y_1$ such that $\ph(x_1,y_1)=1$, +and $x_1$ and $y_1$ are then linearly independent. Define +$$W=\sset{x|\ph(x,x_1) = 0 \tand \ph(x,y_1)=0}\subset V.$$ +That is, $W$ is the kernel of the homomorphism $\ps: V\rtarr F\times F$ specified +by $\ps(x)= (\ph(x,x_1),\ph(x,y_1))$. Since +$\ps(x_1)=(0,1)$ and $\ps(y_1)=(-1,0)$, $\ps$ is an epimorphism. Thus $\dim W = \dim V -2$. +Since $\ph$ restricts to a nonsingular skew symmetric bilinear form on $W$, +the conclusion follows from the induction hypothesis. +\end{proof} + +\section{The index of compact oriented manifolds} + +To study manifolds of dimension $4k$, we consider an analogue for symmetric bilinear +forms of the previous algebraic lemma. Since we will need to take square roots, we +will work over $\bR$. + +\begin{lem} +Let $V$ be a finite dimensional real vector space and $\ph: V\times V \rtarr \bR$ be a +nonsingular symmetric bilinear form. Define $q(x)=\ph(x,x)$. Then $V$ has a basis +$\sset{x_1,\ldots\!,x_r, y_1,\ldots\!,y_s}$ such that $\ph(z,w)=0$ for all +pairs $(z,w)$ of distinct basis elements, $q(x_i)=1$ for $1\leq i\leq r$ +and $q(y_j)=-1$ for $1\leq j\leq s$. The number $r-s$ is an invariant of $\ph$, called the +signature\index{signature} of $\ph$. +\end{lem} +\begin{proof} +We proceed by induction on $\dim V$, and we may assume that $V\neq 0$. Clearly $q(rx)=r^2q(x)$. +Since we can take square roots in $\bR$, we can choose $x_1\in V$ such that $q(x_1)=\pm 1$. Define +$\ps: V\rtarr \bR$ by $\ps(x)=\ph(x,x_1)$ and let $W=\ker \ps$. Since $\ps(x_1)=\pm 1$, $\ps$ +is an epimorphism and $\dim W=\dim V-1$. Since $\ph$ restricts to a nonsingular symmetric bilinear +form on $W$, the existence of a basis as specified follows directly from the induction hypothesis. +Invariance means that the integer $r-s$ is independent of the choice of basis on which +$q$ takes values $\pm 1$, and we leave the verification to the reader. +\end{proof} + +\begin{defn} Let $M$ be a compact oriented $n$-manifold. If $n=4k$, define the index\index{index} +of $M$, denoted $I(M)$, to be the signature of the cup product form +$H^{2k}(M;\bR)\ten H^{2k}(M;\bR)\rtarr \bR$. If $n\,\not\!\equiv\,0 \ \text{mod}\ 4$, define +$I(M)=0$. +\end{defn} + +The Euler characteristic and index are related by the following congruence. + +\begin{prop} For any compact oriented $n$-manifold, $\ch(M)\equiv I(M)\ \text{mod}\ 2$. +\end{prop} +\begin{proof} If $n$ is odd, then $\ch(M)=0$ and $I(M)=0$. If $n\equiv 2\ \text{mod}\ 4$, +then $\ch(M)$ is even and $I(M)=0$. If $n=4k$, then $I(M) = r-s$, +where $r+s = \dim H^{2k}(M;\bR) \equiv \ch(M)\ \text{mod}\ 2$. +\end{proof} + +Observe that the index of $M$ changes sign if the orientation of $M$ is reversed. We +write $-M$ for $M$ with the reversed orientation, and then $I(-M)=-I(M)$. +We also have the following algebraic identities. Write $H^*(M)=H^*(M;\bR)$. + +\begin{lem} If $M$ and $M'$ are compact oriented $n$-manifolds, then +$$I(M\amalg M')=I(M)+I(M'),$$ +where $M\amalg M'$ is given the evident orientation induced from those of $M$ and $M'$. +\end{lem} +\begin{proof} +There is nothing to prove unless $n=4k$, in which case +$$H^{2k}(M\amalg M')=H^{2k}(M)\times H^{2k}(M').$$ +Clearly the cup product of an element of $H^*(M)$ with an element of $H^*(M')$ is zero, and +the cup product form on $H^{2k}(M\amalg M')$ is given by +$$\ph((x,x'),(y,y')) = \ph(x,y)+\ph(x',y')$$ +for $x,y\in H^{2k}(M)$ and $x',y'\in H^{2k}(M')$. The conclusion follows since the signature of +a sum of forms is the sum of the signatures. +\end{proof} + +\begin{lem} Let $M$ be a compact oriented $m$-manifold and $N$ be a compact oriented $n$-manifold. +Then +$$I(M\times N)=I(M)\cdot I(N),$$ +where $M\times N$ is given the orientation induced from those of $M$ and $N$. +\end{lem} +\begin{proof} +We must first make sense of the induced orientation on $M\times N$. For CW pairs $(X,A)$ and $(Y,B)$, +we have an identification of CW complexes +$$ (X\times Y)/(X\times B\cup A\times Y)\iso (X/A)\sma (Y/B)$$ +and therefore an isomorphism +$$ C_*(X\times Y,\, X\times B\cup A\times Y)\iso C_*(X,A)\ten C_*(Y,B).$$ +This implies a relative K\"{u}nneth theorem\index{Kunneth +theorem@K\"unneth theorem!relative} for arbitrary pairs $(X,A)$ and $(Y,B)$. For +subspaces $K\subset M$ and $L\subset N$, +$$(M\times N, M\times N-K\times L) = (M\times N, M\times (N-L) \cup (M-K)\times N).$$ +In particular, for points $x\in M$ and $y\in Y$, +$$(M\times N, M\times N-(x,y)) = (M\times N, M\times (N-y) \cup (M-x)\times N).$$ +Therefore fundamental classes $z_K$ of $M$ at $K$ and $z_L$ of $N$ at $L$ determine a +fundamental class $z_{K\times L}$ of $M\times N$ at $K\times L$. In particular, the image +under $H_m(M)\ten H_n(N)\rtarr H_{m+n}(M\times N)$ of the tensor product of fundamental +classes of $M$ and $N$ is a fundamental class of $M\times N$. + +Turning to the claimed product formula, we see that there is nothing to prove +unless $m+n=4k$, in which case +$$H^{2k}(M\times N)=\sum_{i+j=2k} H^i(M)\ten H^j(N).$$ +The cup product form is given by +$$\ph(x\ten y, x'\ten y') += (-1)^{(\deg y)(\deg x')+ mn}\langle x\cup x',z_M\rangle \langle y\cup y',z_N\rangle$$ +for $x, x'\in H^*(M)$ and $y,y'\in H^*(N)$. If $m$ and $n$ are odd, then the signature +of this form is zero. If $m$ and $n$ are even, then this form is the sum of the tensor +product of the cup product forms on the middle dimensional cohomology groups of $M$ and $N$ +and a form +whose signature is zero. Here, if $m$ and $n$ are congruent to 2 mod 4, the signature +is zero since the lemma of the previous section implies that the signature of the tensor +product of two skew symmetric forms is zero. When $m$ and $n$ are congruent to 0 mod +4, the conclusion holds since the signature of the tensor product of two symmetric forms +is the product of their signatures. We leave the detailed verifications of these algebraic +statements as exercises for the reader. +\end{proof} + +\section{Manifolds with boundary} + +Let $\bH^n=\sset{(x_1,\ldots\!,x_n)|x_n\geq 0}$ be the upper half-plane in $\bR^n$. Recall that an +$n$-manifold with boundary\index{manifold with boundary} is a Hausdorff space $M$ having a +countable basis of open sets +such that every point of $M$ has a neighborhood homeomorphic to an open subset of $\bH^n$. +A point $x$ is an interior point if it has a neighborhood homeomorphic to an open subset +of $\bH^n-\pa \bH^n\iso \bR^n$; otherwise it is a boundary point. It is a fact called +``invariance of domain''\index{invariance of domain} that if $U$ and $V$ are homeomorphic +subspaces of $\bR^n$ and +$U$ is open, then $V$ is open. Therefore, a homeomorphism of an open subspace of +$\bH^n$ onto an open subspace of $\bH^n$ carries boundary points to boundary points. + +We denote the boundary\index{boundary of a manifold} of an $n$-manifold $M$ by $\pa M$. Thus +$M$ is a manifold without boundary if $\pa M$ is empty; $M$ is said to +be closed\index{closed manifold} if, in addition, +it is compact. The space $\pa M$ is an $(n-1)$-manifold without boundary. + +It is a fundamental question in topology to determine which closed manifolds are boundaries. +The question makes sense with varying kinds of extra structure. For example, we can ask whether +or not a smooth (= differentiable) closed manifold is the boundary of a smooth manifold (with +the induced smooth structure). Numerical invariants in algebraic topology give criteria. One +such criterion is given by the following consequence of the Poincar\'e duality theorem. +Remember that $\ch(M)=0$ if $M$ is a closed manifold of odd dimension. + +\begin{prop} If $M=\pa W$, where $W$ is a compact $(2m+1)$-manifold, then $\ch(M)=2\ch (W)$. +\end{prop} +\begin{proof} +The product $W\times I$ is a $(2m+2)$-manifold with +$$\pa(W\times I) = (W\times \sset{0}) \cup (M\times I) \cup (W\times \sset{1}).$$ +Let $U=\pa(W\times I)-(W\times \sset{1})$ and $V=\pa(W\times I)-(W\times \sset{0})$. +Then $U$ and $V$ are open subsets of $\pa(W\times I)$. Clearly $U$ and $V$ +are both homotopy equivalent to $W$ and $U\cap V$ is homotopy equivalent to $M$. +We have the Mayer-Vietoris sequence +$$\diagram + H_{i+1}(U\cup V) \rto \ddouble & H_i(U\cap V)\rto \dto^{\iso} & H_i(U)\oplus H_i(V) +\rto \dto^{\iso} & H_i(U\cup V) \ddouble\\ + H_{i+1}(\pa(W\times I)) \rto & H_i(M)\rto & H_i(W)\oplus H_i(W) +\rto & H_i(\pa(W\times I)).\\ +\enddiagram$$ +Therefore $2\ch(W)=\ch(M)+\ch(\pa(W\times I))$. However, $\ch(\pa(W\times I))=0$ since +$\pa(W\times I)$ is a closed manifold of odd dimension. +\end{proof} + +\begin{cor} If $M=\pa W$ for a compact manifold $W$, then $\ch(M)$ is even. +\end{cor} + +For example, since $\ch(\bR P^{2m})=1$ and $\ch(\bC P^n)=n+1$, this criterion +shows that $\bR P^{2m}$ and $\bC P^{2m}$ cannot be boundaries. Notice that +we have proved that these are not boundaries of topological manifolds, let +alone of smooth ones. + +\section{Poincar\'e duality for manifolds with boundary} + +The index gives a more striking criterion: if a closed oriented $4k$-manifold $M$ is the +boundary of a (topological) manifold, then $I(M)=0$. To prove this, we must first obtain +a relative form of the Poincar\'e duality theorem applicable to manifolds with boundary. + +We let $M$ be an $n$-manifold with boundary, $n>0$, throughout this section, and we let +$R$ be a given commutative ring. We say that $M$ +is $R$-orientable\index{Rorientable@$R$-orientable} +(or orientable\index{orientable} if $R=\bZ$) +if its interior $\cir{M}=M-\pa M$ is $R$-orientable; similarly, an +$R$-orientation\index{Rorientation@$R$-orientation} of $M$ is an +$R$-orientation of its interior. To study these notions, we shall need the following +result, which is intuitively clear but is somewhat technical to prove. In the case of +smooth manifolds, it can be seen in terms of inward-pointing unit vectors of the normal +line bundle of the embedding $\pa M\rtarr M$. + +\begin{thm}[Topological collaring]\index{topological collar} There is an open neighborhood +$V$ of $\pa M$ in $M$ such that the identification $\pa M = \pa M\times \sset{0}$ extends +to a homeomorphism $V\iso \pa M\times [0,1)$. +\end{thm} + +It follows that the inclusion $\cir{M}\rtarr M$ is a homotopy equivalence and the +inclusion $\pa M\rtarr M$ is a cofibration. We take homology with coefficients +in $R$ in the next two results. + +\begin{prop} +An $R$-orientation of $M$ determines an $R$-orientation of $\pa M$. +\end{prop} +\begin{proof} +Consider a coordinate chart $U$ of a point $x\in \pa M$. If $\dim M= n$, then $U$ is +homeomorphic to an open half-disk in $\bH^n$. Let $V=\pa U = U\cap \pa M$ and let +$y\in \cir{U}=U-V$. We have the following chain of isomorphisms: +\begin{eqnarray*} +H_n(\cir{M},\cir{M}-\cir{U}) & \iso & H_n(\cir{M},\cir{M}-y) \\ +& \iso & H_n(M,M-y) \\ +& \iso & H_n(M,M-\cir{U})\\ +& \overto{\pa} & H_{n-1}(M-\cir{U},M-U) \\ +& \iso & H_{n-1}(M-\cir{U},(M-\cir{U})-x)\\ +& \iso & H_{n-1}(\pa M,\pa M-x)\\ +& \iso & H_{n-1}(\pa M,\pa M-V). +\end{eqnarray*} +The first and last isomorphisms are restrictions of the sort that enter into the +definition of an $R$-orientation, and the third isomorphism is similar. We see by +use of a small boundary collar that the inclusion $(\cir{M},\cir{M}-y) \rtarr (M,M-y)$ +is a homotopy equivalence, and that gives the second isomorphism. The connecting +homomorphism is that of the triple $(M,M-\cir{U},M-U)$ and is an isomorphism since +$H_*(M,M-U)\iso H_*(M,M)=0$. The isomorphism that follows comes from the observation that +the inclusion $(M-\cir{U})-x \rtarr M-U$ is a homotopy equivalence, and the next to last +isomorphism is given by excision of $\cir{M}-\cir{U}$. The conclusion is an easy consequence +of these isomorphisms. +\end{proof} + +\begin{prop} If $M$ is compact and $R$-oriented and $z_{\pa M}\in H_{n-1}(\pa M)$ is +the fundamental class determined by the induced $R$-orientation on $\pa M$, then there +is a unique element $z\in H_n(M,\pa M)$ such that $\pa z = z_{\pa M}$; $z$ is called +the $R$-fundamental class\index{Rfundamental class@$R$-fundamental class} determined +by the $R$-orientation of $M$. +\end{prop} +\begin{proof} +Since $\cir{M}$ is a non-compact manifold without boundary and $\cir{M}\rtarr M$ is a +homotopy equivalence, $H_n(M)\iso H_n(\cir{M})=0$ by the vanishing theorem. Therefore +$\pa: H_n(M,\pa M)\rtarr H_{n-1}(\pa M)$ +is a monomorphism. Let $V$ be a +boundary collar and let $N=M-V$. Then $N$ is a closed subspace and a deformation +retract of the $R$-oriented open manifold $\cir{M}$, and we have +$$H_n(\cir{M},\cir{M}-N)\iso H_n(M,M-\cir{M}) = H_n(M,\pa M).$$ +Since $M$ is compact, $N$ is a compact subspace of $\cir{M}$. Therefore the $R$-orientation +of $\cir{M}$ determines a fundamental class in $H_n(\cir{M},\cir{M}-N)$. Let $z$ be its +image in $H_n(M,\pa M)$. Then $z$ restricts to a generator of +$H_n(M,M-y)\iso H_n(\cir{M},\cir{M}-y)$ for every $y\in \cir{M}$. Via naturality diagrams and the +chain of isomorphisms in the previous proof, we see that $\pa z$ restricts to a +generator of $H_{n-1}(\pa M,\pa M-x)$ for all $x\in \pa M$ and is the fundamental +class determined by the $R$-orientation of $\pa M$. +\end{proof} + +\begin{thm}[Relative Poincar\'e duality]\index{Poincare duality theorem@Poincar\'e duality +theorem!relative} Let $M$ be a compact $R$-oriented $n$-\linebreak +manifold +with $R$-fundamental class $z\in H_n(M,\pa M;R)$. Then, with coefficients taken in +any $R$-module $\pi$, capping with $z$ specifies duality isomorphisms +$$ D: H^p(M,\pa M)\rtarr H_{n-p}(M) \ \ \tand \ \ D:H^p(M)\rtarr H_{n-p}(M,\pa M).$$ +\end{thm} +\begin{proof} +The following diagram commutes by inspection of definitions: +$$\diagram +H^{p-1}(\pa M) \rto \dto_D & H^p(M,\pa M) \rto \dto^D +& H^p(M) \rto \dto^D & H^p(\pa M) \dto^D\\ +H_{n-p}(\pa M) \rto & H_{n-p}(M) \rto +& H_{n-p}(M,\pa M) \rto & H_{n-p-1}(\pa M).\\ +\enddiagram$$ +Here $D$ for $\pa M$ is obtained by capping with $\pa z$ and is an isomorphism. +By the five lemma, it suffices to prove that $D: H^p(M)\rtarr H_{n-p}(M,\pa M)$ +is an isomorphism. To this end, let $N = M\cup_{\pa M} M$ be the ``double''\index{double +of a manifold} of +$M$ and let $M_1$ and $M_2$ be the two copies of $M$ in $N$. Clearly $N$ is a +compact manifold without boundary, and it is easy to see that $N$ inherits an +$R$-orientation from the orientation on $M_1$ and the negative of the orientation +on $M_2$. Of course, $\pa M = M_1\cap M_2$. If $U$ is the union of $M_1$ and a +boundary collar in $M_2$ and $V$ is the union of $M_2$ and a boundary collar in +$M_1$, then we have a Mayer-Vietoris sequence for the triad $(N;U,V)$. Using the +evident equivalences of $U$ with $M_1$, $V$ with $M_2$, and $U\cap V$ with $\pa M$, +this gives the exact sequence in the top row of the following commutative diagram. +The bottom row is the exact sequence of the pair $(N,\pa M)$, and the isomorphism +results from the homeomorphism $N/\pa M\iso (M_1/\pa M) \vee (M_2/\pa M)$; we +abbreviate $N_1=(M_1,\pa M)$ and $N_2=(M_2,\pa M)$: +$$\diagram +H^p(N) \rto \dto_D & H^p(M_1) \oplus H^p(M_2) \rto^{\ps} \dto^{D\oplus D} +& H^p(\pa M) \dto^D \rto^{\DE} & H^{p+1}(N) \dto^D \\ +H_{n-p}(N) \rto \ddouble & H_{n-p}(N_1)\oplus H_{n-p}(N_2) \rto \dto^{\iso} +& H_{n-p-1}(\pa M) \ddouble \rto & H_{n-p-1}(N) \ddouble \\ +H_{n-p}(N) \rto & H_{n-p}(N,\pa M) \rto +& H_{n-p-1}(\pa M) \rto & H_{n-p-1}(N). \\ +\enddiagram$$ +The top left square commutes by naturality. In the top middle square, we have +$\ps(x,y)=i_1^*(x)-i_2^*(y)$, where $i_1: \pa M\rtarr M_1$ and $i_2: \pa M\rtarr M_2$ +are the inclusions. Since $D$ for $M_2$ is the negative of $D$ for $M_1$ under the +identifications with $M$, the commutativity of this square follows from the relation +$D\com i^* = \pa\com D: H^p(M)\rtarr H_{n-p-1}(\pa M)$, $i: \pa M\rtarr M$, which holds +by inspection of definitions. For the top right square, $\DE$ is the the top composite +in the diagram +$$\diagram +H^p(\pa M)\rto^(0.3){\de} \dto_D & H^{p+1}(M_1,\pa M)\iso H^{p+1}(N,M_2)\dto^D \rto & H^{p+1}(N)\dto^D \\ +H_{n-p-1}(\pa M) \rto_{{i_1}_*} & H_{n-p-1}(M_1) \rto & H_{n-p-1}(N). +\enddiagram$$ +The right square commutes by naturality, and $D\com \de = {i_1}_*\com D$ by inspection +of definitions. By the five lemma, since the duality maps $D$ for $N$ and $\pa M$ are +isomorphisms, both maps $D$ between direct summands must be isomorphisms. The conclusion follows. +\end{proof} + +\section{The index of manifolds that are boundaries} + +We shall prove the following theorem. + +\begin{thm} If $M$ is the boundary of a compact oriented $(4k+1)$-manifold, then +$I(M)=0$. +\end{thm} + +We first give an algebraic criterion for the vanishing of the signature of a form +and then show that the cup product form on the middle dimensional cohomology of $M$ +satisfies the criterion. + +\begin{lem} Let $W$ be a $n$-dimensional subspace of a $2n$-dimensional real vector +space $V$. Let $\ph: V\times V\rtarr \bR$ be a nonsingular symmetric bilinear form +such that $\ph: W\times W\rtarr \bR$ is identically zero. Then the signature of $\ph$ +is zero. +\end{lem} +\begin{proof} +Let $r$ and $s$ be as in the definition of the signature. Then $r+s=2n$ and we must +show that $r=s$. We prove that $r\geq n$. Applied to the form $-\ph$, this will also +give that $s\geq n$, implying the conclusion. We proceed by induction on $n$. Let +$\sset{x_1,\ldots\!,x_n,z_1,\ldots\!,z_n}$ be a basis for $V$, where $\sset{x_1,\ldots\!,x_n}$ +is a basis for $W$. Define $\tha: V\rtarr \bR^n$ and $\ps: V\rtarr \bR^n$ by +$$\tha(x) = (\ph(x,x_1),\ldots\!,\ph(x,x_n)) \ \tand \ \ps(x) = (\ph(x,z_1),\ldots\!,\ph(x,z_n)).$$ +Since $\ph$ is nonsingular, $\ker{\tha}\cap\ker{\ps}=0$. Since $\ker{\tha}$ and $\ker{\ps}$ each +have dimension at least $n$, neither can have dimension more than $n$ and $\tha$ and $\ps$ must +both be epimorphisms. Choose $y_1$ such that $\tha(y_1)=(1,0,\ldots\!,0)$. Let +$q(x)=\ph(x,x)$ and note that $q(x)=0$ if $x\in W$. Since $q(x_1)=0$ and $\ph(x_1,y_1)=1$, +$q(ax_1+y_1) = 2a+q(y_1)$ for $a\in\bR$. Taking $a=(1-q(y_1))/2$, we find $q(ax_1+y_1)=1$. If +$n=1$, this gives $r\geq 1$ and completes the proof. If $n>1$, define $\om: V\rtarr \bR^2$ by +$\om(x)=(\ph(x,x_1),\ph(x,y_1))$. Since $\om(x_1)=(0,1)$ and $\om(y_1)=(1,q(y_1))$, $\om$ is +an epimorphism. Let $V'=\ker\om$ and let $W'\subset V'$ be the span of $\sset{x_2,\ldots\!,x_n}$. +The restriction of $\ph$ to $V'$ satisfies the hypothesis of the lemma, and the induction +hypothesis together with the construction just given imply that $r\geq n$. +\end{proof} + +Take homology and cohomology with coefficients in $\bR$. + +\begin{lem} Let $M=\pa W$, where $W$ is a compact oriented $(4k+1)$-manifold, +and let $i: M\rtarr W$ be the inclusion. Let +$\ph: H^{2k}(M)\ten H^{2k}(M)\rtarr \bR$ be the cup product form. Then the image +of $i^*: H^{2k}(W)\rtarr H^{2k}(M)$ is a subspace of half the dimension of $H^{2k}(M)$ +on which $\ph$ is identically zero. +\end{lem} +\begin{proof} +Let $z\in H_{4k+1}(W,M)$ be the fundamental class. For $\al,\be\in H^{2k}(W)$, +$$\ph(i^*(\al),i^*(\be))=\langle i^*(\al\cup\be),\pa z\rangle +=\langle \al\cup\be,i_*\pa z\rangle =0$$ +since $i_*\pa=0$ by the long exact sequence of the pair $(W,M)$. Thus $\ph$ is +identically zero on $\im i^*$. The commutative diagram with exact rows +$$\diagram +H^{2k}(W) \rto^{i^*} \dto_D & H^{2k}(M) \rto^{\de} \dto^D & H^{2k+1}(W,M) \dto^D\\ +H_{2k+1}(W,M) \rto_{\pa} & H_{2k}(M) \rto_{i_*} & H_{2k}(W) +\enddiagram$$ +implies that $H^{2k}(M)\iso \im i^*\oplus \im\de\iso \im i^*\oplus \im i_*$. Since +$i^*$ and $i_*$ are dual homomorphisms, $\im i^*$ and $\im i_*$ are dual vector spaces +and thus have the same dimension. +\end{proof} + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} + +Let $M$ be a compact connected $n$-manifold with boundary $\pa M$, where $n\geq 2$. +\begin{enumerate} +\item Prove: $\pa M$ is not a retract of $M$. +\item Prove: if $M$ is contractible, then $\pa M$ has the homology of a sphere. +\item Assume that $M$ is orientable. Let $n = 2m+1$ and let $K$ be the kernel of the +homomorphism $H_m(\pa M) \rtarr H_m(M)$ induced by the inclusion, where homology is taken +with coefficients in a field. Prove: $\dimÊÊ\,H_m(\pa M)Ê=Ê2\dimÊÊ\,K$. +\end{enumerate} + +Let $n = 3$ in the rest of the problems. + +\begin{enumerate} +\item[4.] Prove: if $M$ is orientable, $\pa M$ is empty, and $H_1(M; \bZ) = 0$, then $M$ has +the same homology groups as a $3$-sphere. +\item[5.] Prove: if $M$ is nonorientable and $\pa M$ is empty, then $H_1(M;\bZ)$ is infinite. +\end{enumerate} + +(Hint for the last three problems: use the standard classification of closed $2$-manifolds +and think about first homology groups.) + +\begin{enumerate} +\item[6.] Prove: if $M$ is orientable and $H_1(M;\bZ) = 0$, then $\pa M$ is a disjoint union +of $2$-spheres. +\item[7.] Prove: if $M$ is orientable, $\pa M \neq\ph$, and $\pa M$ contains no $2$-spheres, +then $H_1(M;\bZ)$ is infinite. +\item[8.] Prove: if $M$ is nonorientable and $\pa M$ contains no $2$-spheres and no projective +planes, then $H_1(M;\bZ)$ is infinite. +\end{enumerate} + +\chapter{Homology, cohomology, and $K(\pi,n)$s} + +We have given an axiomatic definition of ordinary homology and cohomology, and we have shown +how to realize the axioms by means of either cellular or singular chain and cochain complexes. +We here give a homotopical way of constructing ordinary theories that makes no use of chains, +whether cellular or singular. We also show how to construct cup and cap products homotopically. +This representation of homology and cohomology in terms of Eilenberg-Mac\,Lane spaces is the +starting point of the modern approach to homology and cohomology theory, and we shall indicate +how theories that do not satisfy the dimension axiom can be represented. We shall also describe +Postnikov systems, which give a way to approximate general (simple) spaces by weakly equivalent +spaces built up out of Eilenberg-Mac\,Lane spaces. This is conceptually dual to the way that CW +complexes allow the approximation of spaces by weakly equivalent spaces built up out of spheres. +Finally, we present the important notion of cohomology operations and +relate them to the cohomology of Eilenberg-Mac\,Lane spaces. + +\section{$K(\pi,n)$s and homology} + +Recall that a reduced homology theory on based CW complexes is a +sequence of functors +$\tilde{E}_q$ from the homotopy category of based CW complexes to the category of Abelian +groups. Each $\tilde{E}_q$ must satisfy the exactness and additivity axioms, and there must +be a natural suspension isomorphism. Up to isomorphism, ordinary reduced homology with +coefficients in $\pi$ is characterized as the unique such theory that satisfies the +dimension axiom: $\tilde{E}_0(S^0) = \pi$ and $\tilde{E}_q(S^0) = 0$ if $q\neq 0$. We +proceed to construct such a theory homotopically. + +For based spaces $X$ and $Y$, we let $[X,Y]$ denote the set of based homotopy classes of based +maps $X\rtarr Y$. Recall that we require Eilenberg-Mac\,Lane spaces $K(\pi,n)$ to have the +homotopy\index{Eilenberg-Mac\,Lane space} types of CW complexes and that, up to homotopy equivalence, there is a unique such +space for each $n$ and $\pi$. By a result of Milnor, if $X$ has the homotopy type of a CW +complex, then so does $\OM X$. By the Whitehead theorem, we therefore have a homotopy equivalence +$$\tilde{\si}: K(\pi,n)\rtarr \OM K(\pi,n+1).$$ +This map is the adjoint of a map +$$\si: \SI K(\pi,n) \rtarr K(\pi,n+1).$$ +We may take the smash product of the map $\si$ with a based CW complex $X$ and use the +suspension homomorphism on homotopy groups to obtain maps +\begin{eqnarray*} +\pi_{q+n}(X\sma K(\pi,n)) & \overto{\SI} & \pi_{q+n+1}(\SI(X\sma K(\pi,n))) \\ + & = & \pi_{q+n+1}(X\sma \SI K(\pi,n)) \overto{(\id\sma\si)_*} +\pi_{q+n+1}(X\sma K(\pi,n+1)). +\end{eqnarray*} + +\begin{thm} +For CW complexes $X$, Abelian groups $\pi$ and integers $n\geq 0$, there are natural +isomorphisms\index{homology theory!ordinary} +$$ \tilde{H}_q(X;\pi)\iso \colim_{n}\pi_{q+n}(X\sma K(\pi,n)).$$ +\end{thm} + +It suffices to verify the axioms, and the dimension axiom is clear. If $X=S^0$, +then $X\sma K(\pi,n)=K(\pi,n)$. Here the homotopy groups in the colimit system are +zero if $q\neq 0$, and, if $q=0$, the colimit runs over a sequence of isomorphisms +between copies of $\pi$. + +The verifications of the rest of the axioms are exercises in the use of the homotopy excision +and Freudenthal suspension theorems, and it is worthwhile to carry out these exercises in +greater generality. + +\begin{defn} A prespectrum\index{prespectrum} is a sequence of based spaces $T_n$, $n\geq 0$, and +based maps $\si: \SI T_n \rtarr T_{n+1}$. +\end{defn} + +The example at hand is the Eilenberg-Mac\,Lane prespectrum $\sset{K(\pi,n)}$. Another +example is the ``suspension prespectrum''\index{suspension prespectrum} $\sset{\SI^n X}$ of +a based space $X$; the required maps $\SI(\SI^nX) \rtarr \SI^{n+1}X$ are the evident identifications. When $X=S^0$, this +is called the sphere prespectrum.\index{sphere prespectrum} + +\begin{thm} Let $\sset{T_n}$ be a prespectrum such that $T_n$ is $(n-1)$-connected and +of the homotopy type of a CW complex for each $n$. Define\index{homology theory!reduced} +$$\tilde{E}_q(X) = \colim_{n}\pi_{q+n}(X\sma T_n),$$ +where the colimit is taken over the maps +$$ +\pi_{q+n}(X\sma T_n) \overto{\SI} \pi_{q+n+1}(\SI(X\sma T_n)) \iso + \pi_{q+n+1}(X\sma \SI T_n) \overto{\id\sma\si} +\pi_{q+n+1}(X\sma T_{n+1}). +$$ +Then the functors $\tilde{E}_q$ define a reduced homology theory on based CW complexes. +\end{thm} +\begin{proof} +Certainly the $\tilde{E}$ are well defined functors from the homotopy category of based +CW complexes to the category of Abelian groups. We must verify the exactness, additivity, +and suspension axioms. Without loss of generality, we may take the $T_n$ to be CW complexes +with one vertex and no other cells of dimension less than $n$. Then $X\sma T_n$ is a quotient +complex of $X\times T_n$, and it too has one vertex and no other cells of dimension less than $n$. +In particular, it is $(n-1)$-connected. + +If $A$ is a subcomplex of $X$, then the homotopy excision theorem implies that the quotient map +$$(X\sma T_n,A\sma T_n) \rtarr ((X\sma T_n)/(A\sma T_n),*)\iso ((X/A)\sma T_n,*)$$ +is a $(2n-1)$-equivalence. We may restrict to terms with $n>q-1$ in calculating $\tilde{E}_q(X)$, +and, for such $q$, the long exact sequence of homotopy groups of the pair $(X\sma T_n,A\sma T_n)$ +gives that the sequence +$$\pi_{q+n}(A\sma T_n) \rtarr \pi_{q+n}(X\sma T_n) \rtarr \pi_{q+n}((X/A)\sma T_n)$$ +is exact. Since passage to colimits preserves exact sequences, this proves the exactness +axiom. + +We need some preliminaries to prove the additivity axiom. + +\begin{defn} Define the weak product\index{weak product} $\prod^{w}_{\, _i} Y_i$ of a set of based +spaces $Y_i$ to be +the subspace of $\prod_i Y_i$ consisting of those points all but finitely many of whose coordinates +are basepoints. +\end{defn} + +\begin{lem} For a set of based spaces $\sset{Y_i}$, the canonical map +$$\textstyle{\sum}_i \pi_q(Y_i) \rtarr \pi_q(\textstyle{\prod}^w_{\, i} Y_i)$$ +is an isomorphism. +\end{lem} +\begin{proof} The homotopy groups of $\prod^w_{\, i} Y_i$ are the colimits of the homotopy +groups of the finite subproducts of the $Y_i$, and the conclusion follows. +\end{proof} + +\begin{lem} If $\sset{Y_i}$ is a set of based CW complexes, then $\prod^w_{\, i}Y_i$ +is a CW complex whose cells are the cells of the finite subproducts of the $Y_i$. +If each $Y_i$ has a single vertex and no $q$-cells for $qn$. The system can be displayed diagrammatically +as follows: +$$\diagram +& \vdots \dto & \\ +& X_{n+1} \dto^{p_{n+1}} \rto^(0.29){k^{n+3}} & K(\pi_{n+2}(X),n+3) \\ +X \urto^{\al_{n+1}} \ddrto^{\al_1} \rto^{\al_n} & X_n \rto^(0.27){k^{n+2}} \dto & K(\pi_{n+1}(X),n+2) \\ +& \vdots \dto & \\ +& X_1 \rto^(0.33){k^3} & K(\pi_{2}(X),3). & \\ +\enddiagram$$ +Our requirement that Eilenberg-Mac\,Lane spaces +have the homotopy types of CW complexes implies (by a result of Milnor) +that each $X_{n}$ has the homotopy type of a CW complex. The maps $\alpha _{n}$ +induce a weak equivalence $X \rightarrow \lim X_{n}$, but the inverse limit +generally will not have the homotopy type of a CW complex. The ``$k$-invariants'' $\sset{k^{n+2}}$ +\index{kinvariants@$k$-invariants} that specify the system are to be regarded as cohomology classes +\[ k^{n+2}\in H^{n+2}(X_{n};{\pi}_{n+1}(X)). \] +These classes together with the homotopy groups $\pi_{n}(X)$ specify the +weak homotopy type of $X$. We outline the proof of the following theorem. + +\begin{thm} +A simple space $X$ of the homotopy type of a CW complex has a +Postnikov system. +\end{thm} +\begin{proof} +Assume inductively that $\alpha _{n}: X \rightarrow X_{n}$ has been constructed. +A consequence of the homotopy excision theorem shows that the cofiber $C(\alpha _{n})$ +is $(n+1)$-connected and satisfies +\[ {\pi} _{n+2}(C(\alpha_{n}))={\pi} _{n+1}(X). \] +More precisely, the canonical map $\et: F(\alpha _{n}) \rightarrow \Omega C(\alpha _{n})$ +induces +an isomorphism on ${\pi} _{q}$ for $q\leq n+1$. We construct +\[ j: C(\alpha _{n}) \rightarrow K({\pi} _{n+1}(X),n+2) \] +by inductively attaching cells to $C(\alpha _{n})$ to kill its higher +homotopy groups. We take the composite of $j$ and the inclusion +$X_{n} \subset C(\alpha _{n})$ to be the $k$-invariant +$$k^{n+2}: X_n \rtarr K(\pi_{n+1}(X),n+2).$$ +By our definition of a Postnikov system, we must define $X_{n+1}$ to be the +homotopy fiber of $k^{n+2}$. Thus its points are pairs $(\omega ,x)$ consisting +of a path $\omega : I\rightarrow K({\pi}_{n+1}(X),n+2)$ and a point $x\in X_{n}$ +such that $\omega (0)=*$ and $\omega (1)=k^{n+2}(x)$. The map +$p_{n+1}: X_{n+1} \rightarrow X_{n}$ is given by +$p_{n+1}(\omega ,x)=x$, and the map $\alpha _{n+1}: X \rightarrow X_{n+1}$ +is given by $\alpha _{n+1}(x)=(\omega (x),\alpha_{n}(x))$, where +$\omega (x)(t) = j(x,1-t)$, $(x,1-t)$ being a point on the cone +$CX \subset C(\alpha _{n})$. Clearly $p_{n+1}\com\alpha _{n+1} = \alpha _{n}$. +It is evident that $\alpha _{n+1}$ induces an isomorphism on +${\pi}_{q}$ for $q\leq n$, and a diagram chase shows that this also holds +for $q=n+1$. +\end{proof} + +\section{Cohomology operations} + +Consider a ``represented functor''\index{represented functor} $k(X)=[X,Z]$ and another +contravariant functor +$k'$ from the homotopy category of based CW complexes to the category of sets. The +following simple observation actually applies to represented functors on arbitrary +categories. We shall use it to describe cohomology operations, but it also applies to +describe many other invariants in algebraic topology, such as the characteristic classes +of vector bundles. + +\begin{lem}[Yoneda]\index{Yoneda lemma} There is a canonical bijection between natural +transformations +$\PH: k\rtarr k'$ and elements $\ph\in k'(Z)$. +\end{lem} +\begin{proof} +Given $\PH$, we define $\ph$ to be $\PH(\id)$, where $\id\in k(Z)=[Z,Z]$ is the +identity map. Given $\ph$, we define $\PH: k(X)\rtarr k'(X)$ by the formula +$\PH(f)=f^*(\ph)$. Here $f$ is a map $X\rtarr Z$, and it induces +$f^*=k'(f):k'(Z)\rtarr k'(X)$. It is simple to check that these are inverse +bijections. +\end{proof} + +We are interested in the case when $k'$ is also represented, say $k'(X)=[X,Z']$. + +\begin{cor} There is a canonical bijection between natural transformations +$\PH: [-,Z]\rtarr [-,Z']$ and elements $\ph\in [Z,Z']$. +\end{cor} + +\begin{defn} Suppose given cohomology theories $\tilde{E}^*$ and $\tilde{F}^*$. A cohomology +operation\index{cohomology operation} of type $q$ and degree $n$ is a natural transformation +$\tilde{E}^q\rtarr \tilde{F}^{q+n}$. A stable cohomology +operation\index{cohomology operation!stable} of degree $n$ is a sequence +$\sset{\PH^q}$ of cohomology operations of type $q$ and degree $n$ such that the following +diagram commutes for each $q$ and each based space $X$: +$$\diagram +\tilde{E}^q(X)\rto^{\PH^q} \dto_{\SI} & \tilde{E}^{q+n}(X) \dto^{\SI} \\ +\tilde{E}^{q+1}(\SI X) \rto_(0.45){\PH^{q+1}} & \tilde{E}^{q+1+n}(\SI X). \\ +\enddiagram$$ +We generally abbreviate notation by setting $\PH^q=\PH$. +\end{defn} + +In general, cohomology operations are only natural transformations of set-valued functors. +However, stable operations are necessarily homomorphisms of cohomology groups, as the +reader is encouraged to check. + +\begin{thm} +Cohomology operations $\tilde{H}^q(-;\pi)\rtarr \tilde{H}^{q+n}(-;\rh)$ are in canonical +bijective correspondence with elements of $\tilde{H}^{q+n}(K(\pi,q);\rh)$. +\end{thm} +\begin{proof} +Translate to the represented level, apply the previous corollary, and translate back. +\end{proof} + +This seems very abstract, but it has very concrete consequences. To determine all cohomology +operations, we need only compute the cohomology of all Eilenberg-Mac\,Lane spaces. We have +described an explicit construction of these spaces as topological Abelian groups +in Chapter 16 \S5, and this construction leads to an inductive method of computation. We briefly +indicate a key example of how this works, without proofs. + +\begin{thm} For $n\geq 0$, there are stable cohomology operations +$$Sq^n: H^q(X;\bZ_2)\rtarr H^{q+n}(X;\bZ_2),$$ +called the Steenrod operations.\index{Steenrod operations} They satisfy the following properties. +\begin{enumerate} +\item[(i)] $Sq^0$ is the identity operation. +\item[(ii)] $Sq^n(x)=x^2$ if $n=\text{\em deg}\,x$ and $Sq^n(x)=0$ if $n> \text{\em deg}\,x$. +\item[(iii)] The Cartan formula\index{Cartan formula} holds: +$$Sq^n(xy)= \sum_{i+j=n}Sq^i(x)Sq^j(y).$$ +\end{enumerate} +\end{thm} + +In fact, the Steenrod operations are uniquely characterized by the stated properties. +There are also formulas, called the Adem relations,\index{Adem relations} describing $Sq^iSq^j$, +as a linear combination of operations $Sq^{i+j-k}Sq^k$, $2k\leq i$, when $0\text{\em dim}\,\xi$. +\item $w_1(\ga_1)\neq 0$, where $\ga_1$ is the universal line bundle over $\bR P^{\infty}$. +\item $w_i(\xi\oplus \epz)= w_i(\xi)$. +\item $w_i(\ze\oplus \xi)= \sum_{j=0}^i w_j(\ze)\cup w_{i-j}(\xi)$. +\end{enumerate} +Every mod $2$ characteristic class for $n$-plane bundles can be written uniquely as a polynomial +in the Stiefel-Whitney classes $\sset{w_1,\ldots\!,w_n}$. +\end{thm} + +\begin{thm} For $n\geq 1$, there are elements $w_i\in H^i(BO(n);\bZ_2)$, $i\geq 0$, called the +Stiefel-Whitney classes. They satisfy and are uniquely characterized by the following +axioms. +\begin{enumerate} +\item $w_0=1$ and $w_i = 0$ if $i>n$. +\item $w_1\neq 0$ when $n=1$. +\item $i_n^*(w_i)=w_i$. +\item $p_{m,n}^*(w_i) = \sum_{j=0}^i w_j\ten w_{i-j}$. +\end{enumerate} +The mod $2$ cohomology $H^*(BO(n);\bZ_2)$ is the polynomial algebra $\bZ_2[w_1,\ldots\!,w_n]$. +\end{thm} + +For the uniqueness, suppose given another collection of classes $w'_i$ for all $n\geq 1$ that +satisfy the stated properties. Since $BO(1)=\bR P^{\infty}$, $w_1=w'_1$ is the unique non-zero +element of $H^1(\bR P^{\infty};\bZ_2)$. Therefore $w_i=w'_i$ for all $i$ when $n=1$, and we +assume that this is true for all $mq-n$. Calculation of $w_i(\nu)$ from the Whitney duality +formula can lead to a contradiction if $q$ is too small. + +One calculation is immediate. Since the normal bundle of the standard embedding +$S^q\rtarr \bR^{q+1}$ is trivial, $w(S^q)=1$. A manifold is said to be +parallelizable\index{parallelizable manifold} if its tangent bundle is trivial. +For some manifolds $M$, we can show that $M$ is not parallelizable by showing that +one of its Stiefel-Whitney classes is non-zero, but this strategy fails for $M=S^q$. + +We describe some standard computations in the cohomology of projective spaces that give +less trivial examples. Write $\ze_q$ for the canonical line bundle\index{canonical line +bundle} over $\bR P^{q}$ in this +section. (We called it $\ga_1^{q+1}$ before.) The total space of $\ze_q$ consists of +pairs $(x,v)$, where $x$ is a line in $\bR^{q+1}$ +and $v$ is a point on that line. This is a subbundle of the trivial $(q+1)$-plane +bundle $\epz^{q+1}$, and we write $\ze_q^{\perp}$ for the complementary bundle whose +points are pairs $(x,w)$ such that $w$ is orthogonal to the line $x$. Thus +$$\ze_q\oplus \ze_q^{\perp}\iso\epz^{q+1}.$$ + +Write $H^*(\bR P^q;\bZ_2) =\bZ_2[\al]/(\al^{q+1})$, +$\deg\al =1$. Thus $\al=w_1(\ze_q)$. Since $\ze_q$ is a line bundle, $w_i(\ze_q)=0$ for +$i>1$. The formula $w(\ze_q)\cup w(\ze_q^{\perp})=1$ implies that +$$w(\ze_q^{\perp}) = 1+\al +\cdots + \al^{q}.$$ + +We can describe $\ta(\bR P^q)$ in terms of $\ze_q$. Consider a point $x\in S^q$ and write +$(x,v)$ for a typical vector in the tangent plane of $S^q$ at $x$. Then $x$ is orthogonal +to $v$ in $\bR^{q+1}$ and $(x,v)$ and $(-x,-v)$ have the same image in $\ta(\bR P^q)$. If +$L_x$ is the line through $x$, then this image point determines and is determined by the +linear map $f: L_x\rtarr L_x^{\perp}$ that sends $x$ to $v$. Starting from this, it is +easy to check that $\ta(\bR P^q)$ is isomorphic to the bundle $\Hom(\ze_q,\ze_q^{\perp})$. +As for any line bundle, we have $\Hom(\ze_q,\ze_q)\iso\epz$ since the identity homomorphisms +of the fibers specify a cross-section. Again, as for any bundle over a smooth manifold, a +choice of Euclidean metric determines an isomorphism $\Hom(\ze_q,\epz)\iso \ze_q$. These +facts give the following calculation of $\ta(\bR P^q)\oplus\epz$: +\begin{eqnarray*} +\ta(\bR P^q)\oplus\epz & \iso & \Hom(\ze_q,\ze_q^{\perp})\oplus\Hom(\ze_q,\ze_q) \\ +& \iso & \Hom(\ze_q,\ze_q^{\perp}\oplus \ze_q) \iso \Hom(\ze_q,\epz^{q+1}) \\ +& \iso & (q+1)\Hom(\ze_q,\epz)\iso (q+1)\ze_q. +\end{eqnarray*} +Therefore +$$w(\bR P^q) = w((q+1)\ze_q) = w(\ze_q)^{q+1} += (1+\al)^{q+1}=\sum_{0\leq i\leq q} + \left(\begin{array}{c}q+1\\i\end{array}\right) \al^i.$$ +Explicit computations are obtained by computing mod $2$ binomial coefficients. + +For example, $w(\bR P^q)=1$ if and only if $q=2^k-1$ for some $k$ (as the reader should +check) and therefore $\bR P^q$ can be parallelizable only if $q$ is of this form. If $\bR^{q+1}$ +admits a bilinear product without zero divisors, then it is not hard to prove that +$\ta(\bR P^{q})\iso \Hom(\ze_q,\ze_q^{\perp})$ admits $q$ linearly independent cross-sections +and is therefore trivial. We conclude that $\bR^{q+1}$ can admit such a product only if $q+1=2^k$ +for some $k$. The real numbers, complex numbers, quaternions, and Cayley numbers show that there +is such a product for $q+1=1$, $2$, $4$, and $8$. As we shall explain in the next chapter, these +are in fact the only $q$ for which $\bR^{q+1}$ admits such a product. + +While the calculation of $w(\bR P^q)$ just given is quite special, there is a remarkable general +recipe, called the ``Wu formula,'' for the computation of $w(M)$ in terms of Poincar\'e duality +and the Steenrod operations in $H^*(M;\bZ_2)$. In analogy with $w(M)$, we define the total +Steenrod square of an element $x$ by $Sq(x)=\sum_i Sq^i(x)$.\index{total Steenrod operation} + +\begin{thm}[Wu formula]\index{Wu formula} Let $M$ be a smooth closed $n$-manifold with +fundamental class +$z\in H_n(M;\bZ_2)$. Then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where +$v=\sum v_i\in H^{**}(M;\bZ_2)$ is the unique cohomology class such that +$$\langle v\cup x,z\rangle = \langle Sq(x),z \rangle $$ +for all $x\in H^*(M;\bZ_2)$. Thus, for $k\geq 0$, $v_k\cup x = Sq^k(x)$ for all +$x\in H^{n-k}(M;\bZ_2)$, and +$$w_k(M)=\sum_{i+j=k}Sq^i(v_j).$$ +\end{thm} + +Here the existence and uniqueness of $v$ is an easy exercise from the Poincar\'e duality +theorem. The basic reason that such a formula holds is that the Stiefel-Whitney classes +can be defined in terms of the Steenrod operations, as we shall see shortly. The Wu formula +implies that the Stiefel-Whitney classes are homotopy invariant: if $f:M\rtarr M'$ is a +homotopy equivalence between smooth closed $n$-manifolds, then +$f^*: H^*(M';\bZ_2)\rtarr H^*(M;\bZ_2)$ satisfies $f^*(w(M'))=w(M)$. In fact, the conclusion +holds for any map $f$, not necessarily a homotopy equivalence, that induces an isomorphism in +mod $2$ cohomology. Since the tangent bundle of $M$ depends on its smooth structure, this +is rather surprising. + +\section{Characteristic numbers of manifolds} + +Characteristic classes determine important numerical invariants of manifolds, called their +characteristic numbers. + +\begin{defn} Let $M$ be a smooth closed $R$-oriented $n$-manifold with fundamental class +$z\in H_n(M;R)$. For a characteristic class $c$ of degree $n$, define the tangential +characteristic number\index{characteristic class}\index{characteristic number!tangential} +\index{characteristic number!normal} $c[M]\in R$ by $c[M] = \langle c(\ta(M)),z \rangle$. +Similarly, define the normal characteristic number $c[\nu(M)]$ by +$c[\nu(M)] = \langle c(\nu(M)),z \rangle$, where $\nu(M)$ is the normal bundle associated +to an embedding of $M$ in $\bR^q$ for $q$ sufficiently large. (These numbers are well defined +because any two embeddings of $M$ in $\bR^q$ for large $q$ are isotopic and have equivalent +normal bundles.) +\end{defn} + +In particular, if $r_i$ are integers such that $\sum ir_i=n$, then the monomial +$w_1^{r_1}\cdots w_n^{r_n}$ is a characteristic class of degree $n$, and all mod $2$ +characteristic classes of degree $n$ are linear combinations of these. Different +manifolds can have the same Stiefel-Whitney numbers.\index{Stiefel-Whitney numbers} In fact, +we have the following observation. + +\begin{lem} If $M$ is the boundary of a smooth compact $(n+1)$-manifold $W$, then +all tangential Stiefel-Whitney numbers of $M$ are zero. +\end{lem} +\begin{proof} +Using a smooth tubular neighborhood, we see that there is an inward-pointing normal +vector field along $M$ that spans a trivial bundle $\epz$ such that +$$\ta(W)|_M\iso \ta(M)\oplus \epz.$$ +Therefore, if $i:M\rtarr W$ is the inclusion, then $i^*(w_j(W))=w_j(M)$. Let $f$ be +a polynomial in the $w_j$ of degree $n$. Recall that the fundamental class of $M$ +is $\pa z$, where $z\in H_{n+1}(W,M)$ is the fundamental class of the pair $(W,M)$. We have +$$\langle f(M),\pa z \rangle += \langle i^*f(W),\pa z \rangle += \langle f(W), i_*\pa z \rangle = 0$$ +since $i_*\pa = 0$ by the long exact homology sequence of the pair. +\end{proof} + +\begin{lem} +All tangential Stiefel-Whitney numbers\index{Stiefel-Whitney numbers!tangential} of a +smooth closed manifold $M$ are zero if +and only if all normal Stiefel-Whitney numbers\index{Stiefel-Whitney numbers!normal} of +$M$ are zero. +\end{lem} +\begin{proof} The Whitney duality formula implies that every $w_i(M)$ is a polynomial +in the $w_i(\nu(M))$ and every $w_i(\nu(M))$ is a polynomial in the $w_i(M)$. +\end{proof} + +We shall explain the following amazing result of Thom in the last chapter. + +\begin{thm}[Thom] If $M$ is a smooth closed $n$-manifold all of whose normal +Stiefel-Whitney numbers are zero, then $M$ is the boundary of a smooth +$(n+1)$-manifold. +\end{thm} + +Thus we need only compute the Stiefel-Whitney numbers of $M$ to determine whether +or not it is a boundary. By Wu's formula, the computation only requires knowledge +of the mod $2$ cohomology of $M$, with its Steenrod operations. In practice, it might +be fiendishly difficult to actually construct a manifold with +boundary $M$ geometrically. + +\section{Thom spaces and the Thom isomorphism theorem} + +There are several ways to construct the Stiefel-Whitney classes. The most illuminating +one depends on a simple, but fundamentally important, construction on vector bundles, +namely their ``Thom spaces.'' This construction will also be at the heart of the proof of +Thom's theorem in the last chapter. + +\begin{defn} Let $\xi: E\rtarr B$ be an $n$-plane bundle. Apply one-point compactification to each +fiber of $\xi$ to obtain a new bundle $Sph(E)$\index{Sph(E)@$Sph(E)$} over $B$ whose fibers are +spheres $S^n$ with +given basepoints, namely the points at $\infty$. These basepoints specify a cross-section +$B\rtarr Sph(E)$. Define the Thom space\index{Thom space} $T\xi$ to be the quotient space +$T(\xi)= Sph(E)/B$. That is, $T(\xi)$ is obtained from $E$ by applying fiberwise one-point +compactification and then identifying all of the points at $\infty$ to a single basepoint +(denoted $\infty$). Observe that this construction is functorial with respect to maps of +vector bundles. +\end{defn} + +\begin{rem} If we give the bundle $\xi$ a Euclidean metric and let $D(E)$ and $S(E)$ +denote its unit disk bundle and unit sphere bundle, then there is an evident +homeomorphism between $T\xi$ and the quotient space $D(E)/S(E)$. In turn, $D(E)/S(E)$ +is homotopy equivalent to the cofiber of the inclusion $S(E)\rtarr D(E)$ and therefore +to the cofiber of the projection $S(E)\rtarr B$. +\end{rem} + +If the bundle $\xi$ is trivial, so that $E=B\times \bR^n$, then $ Sph(E)=B\times S^n$. +Quotienting out $B$ amounts to the same thing as giving $B$ a disjoint basepoint and then +forming the smash product $B_+\sma S^n$. That is, in this case the Thom complex is $\SI^nB_+$. +Therefore, for any cohomology theory $k^*$, +$$k^q(B)=\tilde{k}^q(B_+) \iso \tilde{k}^{n+q}(T\xi).$$ +There is a conceptual way of realizing this isomorphism. For any $n$-plane bundle $\xi: E\rtarr B$, +we have a projection $\xi: Sph(E)\rtarr B$ and a quotient map $\pi: Sph(E)\rtarr T\xi$. +We can compose their product with the diagonal map of $Sph(E)$ to obtain a composite map +$$ Sph(E)\rtarr Sph(E)\times Sph(E) \rtarr B\times T\xi.$$ +This sends all points at $\infty$ to points of $B\times \sset{\infty}$. Therefore it factors through +a map +$$ \DE: T\xi\rtarr B_+\sma T\xi,$$ +which is called the ``Thom diagonal.''\index{Thom diagonal} For a commutative ring $R$, we can +use $\DE$ to define a cup product +$$ H^p(B;R)\ten \tilde{H}^q(T\xi;R) \rtarr \tilde{H}^{p+q}(T\xi;R).$$ +When the bundle $\xi$ is trivial, we let $\mu\in \tilde{H}^n(B_+\sma S^n;R)$ be the suspension of +the identity element $1\in H^0(B;R)$, and we find that $x\rtarr x\cup \mu$ specifies the +suspension isomorphism $H^q(B;R)\iso \tilde{H}^{n+q}(B_+\sma S^n;R) = \tilde{H}^{n+q}(T\xi;R)$. + +Now consider a general bundle $\xi$. On neighborhoods $U$ of $B$ over which $\xi$ is trivial, +we have $H^q(U;R)\iso \tilde{H}^{n+q}(T(\xi|_U);R)$. The isomorphism depends on the trivialization +$\ph_U: U\times \bR^n\rtarr \xi^{-1}(U)$. It is natural to ask if these isomorphisms patch +together to give a global isomorphism $H^q(B_+)\rtarr \tilde{H}^{n+q}(T\xi)$. This should look +very similar to the problem of patching local fundamental classes to obtain a global one; that +is, it looks like a question of orientation. This leads to the following definition and theorem. +For a point $b\in B$, let $S^n_b$ be the one-point compactification of the fiber $\xi^{-1}(b)$; +since $S^n_b$ is the Thom space of $\xi|_b$, we have a canonical map $i_b: S^n_b\rtarr T\xi$. + +\begin{defn} Let $\xi: E\rtarr B$ be an $n$-plane bundle. +An $R$-orientation,\index{Rorientation@$R$-orientation} or Thom class,\index{Thom class} of +$\xi$ is an element $\mu\in \tilde{H}^n(T\xi;R)$ such that, for every point $b\in B$, +$i_b^*(\mu)$ is a generator of the free $R$-module $\tilde{H}^n(S^n_b)$. +\end{defn} + +We leave it as an instructive exercise to verify that an $R$-orientation of a closed $n$-manifold +$M$ determines and is determined by an $R$-orientation of its tangent bundle $\ta(M)$. + +\begin{thm}[Thom isomorphism theorem]\index{Thom isomorphism} Let $\mu\in \tilde{H}^n(T\xi;R)$ +be a Thom class for an $n$-plane bundle $\xi: E\rtarr B$. Define +$$\PH: H^q(B;R)\rtarr \tilde{H}^{n+q}(T\xi;R)$$ +by $\PH (x)=x\cup \mu$. Then $\PH$ is an isomorphism. +\end{thm} +\begin{proof}[Sketch Proof] When $R$ is a field, this can be proved by an inductive Mayer-Vietoris +sequence argument. To exploit inverse images of open subsets of $B$, it is convenient to observe +that, by easy homotopy and excision arguments, +$$\tilde{H}^*(T\xi)\iso H^*(Sph(E),B)\iso H^*(Sph(E),Sph(E)_0)\iso H^*(E,E_0),$$ +where $E_0$ and $Sph(E)_0$ are the subspaces of $E$ and $Sph(E)$ obtained by deleting $\sset{0}$ +from each fiber. Use of a field ensures that the cohomology of the relevant direct limits is the +inverse limit of the cohomologies. An +alternative argument that works for general $R$ can be obtained by first showing that one can +assume that $B$ is a CW complex, by replacing $\xi$ by its pullback along a CW approximation of $B$, +and then proceeding by induction over the restrictions of $\xi$ to the skeleta of $B$; one point +is that the restriction of $\xi$ to any cell is trivial and another is that the cohomology +of $B$ is the inverse limit of the cohomologies of its skeleta. However, much the best proof +from the point of view of anyone seriously interested in algebraic topology is to apply the +Serre spectral sequence of the bundle $Sph(E)$. The Serre spectral sequence\index{Serre spectral +sequence} is a device for +computing the cohomology of the total space $E$ of a fibration from the cohomologies of its base +$B$ and fiber $F$. It measures the cohomological deviation of $H^*(E)$ from $H^*(B)\ten H^*(F)$. +In the present situation, the existence of a Thom class ensures that there is no deviation for +the sphere bundle $Sph(E)\rtarr B$, so that +$$H^*(Sph(E);R)\iso H^*(B;R)\ten H^*(S^n;R).$$ +The section given by the points at $\infty$ induces an isomorphism of $H^*(B;R)\ten H^0(S^n;R)$ +with $H^*(B;R)$, and the quotient map $Sph(E)\rtarr T\xi$ induces an isomorphism of +$\tilde{H}^*(T\xi;R)$ with $H^*(B;R)\ten H^n(S^n;R)$. +\end{proof} + +Just as in orientation theory for manifolds, the question of orientability depends on the +structure of the units of the ring $R$, and this leads to the following conclusion. + +\begin{prop} Every vector bundle admits a unique $\bZ_2$-orientation. +\end{prop} + +This can be proved along with the Thom isomorphism theorem by a Mayer-Vietoris argument. + +\section{The construction of the Stiefel-Whitney classes} + +We indicate two constructions of the Stiefel-Whitney classes. Each has distinct advantages over +the other. First, taking the characteristic class point of view, we define the Stiefel-Whitney +classes\index{Stiefel-Whitney classes} in terms of the Steenrod operations by setting +$$w_i(\xi) = \PH^{-1}Sq^i\PH(1) = \PH^{-1}Sq^i\mu.$$ +Naturality is obvious. Axiom 1 is immediate from the relations $Sq^0=\id$ and +$Sq^i(x)=0$ if $i> \deg\,x$. For axiom 2, we use the following observation. + +\begin{lem} There is a homotopy equivalence $j: \bR P^{\infty}\rtarr T\ga_1$. +\end{lem} +\begin{proof} +$T\ga_1$ is homeomorphic to $D(\ga_1)/S(\ga_1)$. Here $S(\ga_1)$ is the infinite sphere +$S^{\infty}$, which is the universal cover of $\bR P^{\infty}$ and is therefore +contractible. The zero section $\bR P^{\infty}\rtarr D(\ga_1)$ and the quotient map +$D(\ga_1)\rtarr T\ga_1$ are homotopy equivalences, and their composite is the required +homotopy equivalence $j$. +\end{proof} + +Since $Sq^1(x)=x^2$ if $\deg\,x=1$, the lemma implies that $Sq^1$ is non-zero on the Thom class +of $\ga_1$, verifying axiom 2. For axiom 3, we easily check that +$T(\xi\oplus\epz)\iso \SI T(\xi)$ for any vector bundle $\xi$ and that the Thom class of +$\xi\oplus\epz$ is the suspension of the Thom class of $\xi$. Thus axiom 3 follows from the +stability of the Steenrod operations. For axiom 4, we easily check that, for any vector bundles +$\ze$ and $\xi$, $T(\ze\times \xi)\iso T\ze\sma T\xi$ and the Thom class of $\ze\times \xi$ is the +tensor product of the Thom classes of $\ze$ and $\xi$. Interpreting the Cartan formula for the +Steenrod operations externally in the cohomology of products and therefore of smash products, +we see that it implies axiom 4. That is, the properties that axiomatize the Steenrod operations +directly imply the properties that axiomatize the Stiefel-Whitney classes. + +We next take the classifying space point of view. As we shall explain in \S8, +passage from topological groups to their classifying spaces is a product-preserving +functor, at least up to homotopy. We may embed $(\bZ_2)^n = O(1)^n$ in $O(n)$ as the +subgroup of diagonal matrices. The classifying space $BO(1)$ is $\bR P^{\infty}$, +and we obtain a map +$$\om: (\bR P^{\infty})^n \htp B(O(1)^n) \rtarr BO(n) $$ +upon passage to classifying spaces. The symmetric group $\SI_n$ is contained in $O(n)$ +as the subgroup of permutation matrices, and the diagonal subgroup $O(1)^n$ is closed +under conjugation by symmetric matrices. Application of the classifying space functor +to conjugation by permutation matrices induces the corresponding permutation of the factors +of $BO(1)^n$, and it induces the identity map on $BO(n)$. Indeed, up to homotopy, inner +conjugation by an element of $G$ induces the identity map on $BG$ for any topological +group $G$. + +By the K\"unneth theorem, we see that +$$H^*((\bR P^{\infty})^n;\bZ_2) += \ten_{i=1}^n H^*(\bR P^{\infty};\bZ_2) =\bZ_2[\al_1,\ldots\!,\al_n],$$ +where the generators $\al_i$ are of degree one. The symmetric group $\SI_n$ acts on this +cohomology ring by permuting the variables $\al_i$. The subring +$H^*((\bR P^{\infty})^n;\bZ_2)^{\SI_n}$ of elements invariant under +the action is the polynomial algebra on the elementary symmetric functions +$\si_i$, $1\leq i\leq n$, in the variables $\al_i$. Here +$$\si_i = \textstyle{\sum} \al_{j_1}\cdots\al_{j_i},\ \ 1\leq j_1 < \cdots < j_n,$$ +has degree $i$. The induced map $\om^*: H^*(BO(n);\bZ_2)\rtarr H^*((\bR P^{\infty})^n;\bZ_2)$ +takes values in $H^*((\bR P^{\infty})^n;\bZ_2)^{\SI_n}$. We shall give a general reason why +this is so in \S8. The resulting map +$$\om^*: H^*(BO(n);\bZ_2)\rtarr H^*((\bR P^{\infty})^n;\bZ_2)^{\SI_n}$$ +is a ring homomorphism between polynomial algebras on generators of the same degrees. It +turns out to be a monomorphism and therefore an isomorphism. We redefine the Stiefel-Whitney +classes by letting $w_i$ be the unique element such that $\om^*(w_i)=\si_i$ for $1\leq i\leq n$ +and defining $w_0=1$ and $w_i=0$ for $i>n$. Then axioms 1 and 2 for the Stiefel-Whitney +classes are obvious, and we derive axioms 3 and 4 from algebraic properties of elementary +symmetric functions. + +One advantage of this approach is that, since we know the Steenrod +operations on $H^*(\bR P^{\infty};\bZ_2)$ and can read them off on $H^*((\bR P^{\infty})^n;\bZ_2)$ +by the Cartan formula, it leads to a purely algebraic calculation of the Steenrod +operations in $H^*(BO(n);\bZ_2)$. Explicitly, the following ``Wu formula''\index{Wu formula} holds: +$$ Sq^i(w_j) = \sum_{t=0}^i\left(\begin{array}{c}j+t-i-1\\t\end{array}\right) w_{i-t}w_{j+t}.$$ + +\section{Chern, Pontryagin, and Euler classes} + +The theory of the previous sections extends appropriately to complex vector bundles +and to oriented real vector bundles. The proof of the classification theorem for complex +$n$-plane bundles works in exactly the same way as for real $n$-plane bundles, using +complex Grassmann varieties. For oriented real $n$-plane bundles, we use the Grassmann +varieties\index{Grassmann variety!of oriented $n$-planes} of oriented $n$-planes, +the points of which are planes $x$ together with a chosen +orientation. In fact, the fundamental groups of the real Grassmann varieties are $\bZ_2$, +and their universal covers are their orientation covers. These covers are the oriented +Grassmann varieties $\tilde{G}_n(\bR^q)$. We write $BU(n) = G_n(\bC^{\infty})$\index{BUn@$BU(n)$} +and $BSO(n) = \tilde{G}_n(\bR^{\infty})$,\index{BSOn@$BSO(n)$} and we construct universal +complex $n$-plane bundles\index{universal n-plane bundle@universal $n$-plane bundle!complex} +\index{universal n-plane bundle@universal $n$-plane bundle!oriented} +$\ga_n: EU_n\rtarr BU(n)$ and oriented $n$-plane bundles $\tilde{\ga}_n: \tilde{E}_n\rtarr BSO(n)$ +as in the first section. Let $\sE U_n(B)$\index{EUkn(-)@$\sE U_n(B)$} denote the set of equivalence +classes of complex $n$-plane +bundles over $B$ and let $\tilde{\sE}_n(B)$\index{EanBa@$\tilde{\sE}_n(B)$} denote the set of +equivalence classes of oriented +real $n$-plane bundles over $B$; it is required that bundle maps $(g,f)$ be orientation preserving, +in the sense that the induced map of Thom spaces carries the orientation of the target bundle to the +orientation of the source bundle. The universal bundle $\tilde{\ga_n}$ has a canonical orientation +which determines an orientation on $f^*\tilde{E}_n$ for any map $f: B\rtarr BSO(n)$. + +\begin{thm}\index{classification theorem!for complex $n$-plane bundles} The natural +transformation $\PH: [-,BU(n)]\rtarr \sE U_n(-)$ obtained by sending the +homotopy class of a map $f: B\rtarr BU(n)$ to the equivalence class of the $n$-plane +bundle $f^*EU_n$ is a natural isomorphism of functors. +\end{thm} + +\begin{thm}\index{classification theorem!for oriented $n$-plane bundles} The natural +transformation $\PH: [-,BSO(n)]\rtarr \tilde{\sE}_n(-)$ obtained by +sending the homotopy class of a map $f: B\rtarr BSO(n)$ to the equivalence class of the +oriented $n$-plane bundle $f^*\tilde{E}_n$ is a natural isomorphism of functors. +\end{thm} + +The definition of characteristic classes for complex $n$-plane bundles and for oriented +real $n$-plane bundles in a cohomology theory $k^*$ is the same as for real $n$-plane bundles, +and the Yoneda lemma applies. + +\begin{lem} Evaluation on $\ga_n$ specifies a canonical bijection between characteristic +classes of complex $n$-plane bundles and elements of $k^*(BU(n))$. +\end{lem} + +\begin{lem} Evaluation on $\tilde{\ga}_n$ specifies a canonical bijection between characteristic +classes of oriented $n$-plane bundles and elements of $k^*(BSO(n))$. +\end{lem} + +Clearly we have a $2$-fold cover $\pi_n: BSO(n)\rtarr BO(n)$. The mod $2$ characteristic +classes for oriented $n$-plane bundles are as one might expect from this. Continue to write +$w_i$ for $\pi^*(w_i)\in H^i(BSO(n);\bZ_2)$; here $w_1=0$ since $BSO(n)$ is simply connected. + +\begin{thm} $H^*(BSO(n);\bZ_2) \iso \bZ_2[w_2,\ldots\!,w_n]$. +\end{thm} + +If we regard a complex $n$-plane bundle as a real $2n$-plane bundle, then the complex structure +induces a canonical orientation. By the Yoneda lemma, the resulting natural transformation +$r: \sE U_n(-)\rtarr \tilde{\sE}_n(-)$ is represented by a map $r: BU(n)\rtarr BSO(2n)$. Explicitly, ignoring its complex structure, we may identify +$\bC^{\infty}$ with $\bR^{\infty}\oplus\bR^{\infty}\iso \bR^{\infty}$ and so regard a complex +$n$-plane in $\bC^{\infty}$ as +an oriented $2n$-plane in $\bR^{\infty}$. Similarly, we may complexify real bundles fiberwise +and so obtain a natural transformation $c: \sE_n(-)\rtarr \sE U_n(-)$. It is represented by a +map $c: BO(n)\rtarr BU(n)$. Explicitly, identifying $\bC^{\infty}$ with +$\bR^{\infty}\ten_{\bR}{\bC}$, we may complexify an $n$-plane in $\bR^{\infty}$ to obtain an +$n$-plane in $\bC^{\infty}$. + +The Thom space\index{Thom space!of a complex bundle} of a complex or oriented real vector bundle +is the Thom space of its underlying real vector bundle. We obtain characteristic classes in +cohomology with any coefficients by +applying cohomology operations to Thom classes, but it is rarely the case that the resulting +characteristic classes generate all characteristic classes: the cases $H^*(BO(n);\bZ_2)$ and +$H^*(BSO(n);\bZ_2)$ are exceptional. Characteristic classes constructed in this fashion satisfy +homotopy invariance properties that fail for general characteristic classes. + +In the complex case, with integral coefficients, we have a parallel to our second +approach to Stiefel-Whitney classes that leads to a description of $H^*(BU(n);\bZ)$ in +terms of Chern classes. We may embed $(S^1)^n = U(1)^n$ in $U(n)$ as the +subgroup of diagonal matrices. The classifying space $BU(1)$ is $\bC P^{\infty}$, +and we obtain a map +$$\om: (\bC P^{\infty})^n \htp B(U(1)^n) \rtarr BU(n) $$ +upon passage to classifying spaces. The symmetric group $\SI_n$ is contained in $U(n)$ +as the subgroup of permutation matrices, and the diagonal subgroup $U(1)^n$ is closed +under conjugation by symmetric matrices. Application of the classifying space functor +to conjugation by permutation matrices induces the corresponding permutation of the factors +of $BU(1)^n$, and it induces the identity map on $BU(n)$. + +By the K\"unneth theorem, we see that +$$H^*((\bC P^{\infty})^n;\bZ) += \ten_{i=1}^n H^*(\bC P^{\infty};\bZ) =\bZ[\be_1,\ldots\!,\be_n],$$ +where the generators $\be_i$ are of degree two. The symmetric group $\SI_n$ acts on this +cohomology ring by permuting the variables $\be_i$. The subring +$H^*((\bC P^{\infty})^n;\bZ)^{\SI_n}$ of elements invariant under +the action is the polynomial algebra on the elementary symmetric functions +$\si_i$, $1\leq i\leq n$, in the variables $\be_i$. Here +$$\si_i = \textstyle{\sum} \be_{j_1}\cdots\be_{j_i},\ \ 1\leq j_1 < \cdots < j_n,$$ +has degree $2i$. The induced map $\om^*: H^*(BU(n);\bZ)\rtarr H^*((\bC P^{\infty})^n;\bZ)$ +takes values in $H^*((\bC P^{\infty})^n;\bZ)^{\SI_n}$. The resulting map +$$\om^*: H^*(BU(n);\bZ)\rtarr H^*((\bC P^{\infty})^n;\bZ)^{\SI_n}$$ +is a ring homomorphism between polynomial algebras on generators of the same degrees. It +turns out to be a monomorphism and thus an isomorphism when tensored with any field, and it is +therefore an isomorphism. We define the Chern classes by letting $c_i$, $1\leq i\leq n$, be the +unique element such that $\om^*(c_i)=\si_i$. + +\begin{thm} For $n\geq 1$, there are elements $c_i\in H^{2i}(BU(n);\bZ)$, $i\geq 0$, called the +Chern classes.\index{Chern classes} They satisfy and are uniquely characterized by the following +axioms. +\begin{enumerate} +\item $c_0=1$ and $c_i = 0$ if $i>n$. +\item $c_1$ is the canonical generator of $H^2(BU(1);\bZ)$ when $n=1$. +\item $i_n^*(c_i)=c_i$. +\item $p_{m,n}^*(c_i) = \sum_{j=0}^i c_j\ten c_{i-j}$. +\end{enumerate} +The integral cohomology $H^*(BU(n);\bZ)$ is the polynomial algebra $\bZ[c_1,\ldots\!,c_n]$. +\end{thm} + +Here we take axiom 1 as a definition and we interpret axiom 2 as meaning that $c_1$ +corresponds to the identity map of $\bC P^{\infty}$ under the canonical identification of +$[\bC P^{\infty},\bC P^{\infty}]$ with $H^2(\bC P^{\infty};\bZ)$. Axioms 3 and 4 can +be read off from algebraic properties of elementary symmetric functions. The theorem admits +an immediate interpretation in terms of characteristic classes. Observe that, since $H^*(BU(n);\bZ)$ +is a free Abelian group, the theorem remains true precisely as stated with $\bZ$ replaced by any +other commutative ring of coefficients $R$. We continue to write $c_i$ for the image of $c_i$ +in $H^*(BU(n);R)$ under the homomorphism induced by the unit $\bZ\rtarr R$ of the ring $R$. + +The reader deserves to be warned about a basic inconsistency in the literature. + +\begin{rem} With the discussion above, $c_1(\ga_1^{n+1})$ is the canonical generator of +$H^2(\bC P^n;\bZ)$, where $\ga_1^{n+1}$ is the canonical line bundle +\index{canonical line bundle} of lines in $\bC^{n+1}$ +and points on the line. This is the standard convention in algebraic topology. In algebraic +geometry, it is more usual to define Chern classes so that the first Chern class of the dual +of $\ga_1^{n+1}$ is the canonical generator of $H^2(\bC P^n;\bZ)$. With this convention, the +$n$th Chern class would be $(-1)^nc_n$. It is often unclear in the literature which convention +is being followed. +\end{rem} + +Turning to oriented real vector bundles, we define the Pontryagin and Euler classes as +follows, taking cohomology with coefficients in any commutative ring $R$. + +\begin{defn} Define the Pontryagin classes\index{Pontryagin classes} $p_i\in H^{4i}(BO(n);R)$ by +$$p_i = (-1)^ic^*(c_{2i}),$$ +$c^*: H^{4i}(BU(n);R)\rtarr H^{4i}(BO(n);R)$; also write $p_i$ for +$\pi_n^*(p_i)\in H^{4i}(BSO(n);R)$. +\end{defn} + +\begin{defn} Define the Euler class\index{Euler class} $e(\xi)\in H^n(B;R)$ of an $R$-oriented +$n$-plane bundle +$\xi$ over the base space $B$ by $e(\xi)=\PH^{-1}\mu^2$, where $\mu\in H^n(T\xi;R)$ is the Thom +class. Giving the universal oriented $n$-plane bundle over $BSO(n)$ the $R$-orientation +induced by its integral orientation, this defines the Euler class $e\in H^n(BSO(n);R)$. +\end{defn} + +If $n$ is odd, then $2\mu^2=0$ and thus $2e=0$. If $R=\bZ_2$, then $Sq^n(\mu)=\mu^2$ and +thus $e=w_n$. The name ``Euler class'' is justified by the following classical result, +which well illustrates the kind of information that characteristic numbers can +encode.\footnote{See Corollary 11.12 of Milnor and Stasheff {\em Characteristic Classes} +for a proof.} + +\begin{thm} If $M$ is a smooth closed oriented manifold, then the characteristic number +$e[M]=\langle e(\ta(M)),z\rangle\in \bZ$ is the Euler characteristic\index{Euler characteristic} +of $M$. +\end{thm} + +The evident inclusion $T^n \iso SO(2)^n\rtarr SO(2n)$ is a maximal torus, and it induces +a map $BT^n\rtarr BSO(2n)$. A calculation shows that $e$ restricts to the $n$th +elementary symmetric polynomial $\be_1\cdots\be_n$. The cited inclusion factors through +the homomorphism $U(n)\rtarr SO(2n)$, hence $BT^n\rtarr BSO(2n)$ factors through +$r: BU(n)\rtarr BSO(2n)$. This implies another basic fact about the Euler class. + +\begin{prop} $r^*: H^*(BSO(2n);\bZ)\rtarr H^*(BU(n);\bZ)$ sends $e$ to $c_n$. +\end{prop} + +The presence of $2$-torsion makes the description of the integral cohomology rings of $BO(n)$ +and $BSO(n)$ quite complicated, and these rings are almost never used in applications. +Rather, one uses the mod $2$ cohomology rings and the following description of the cohomology +rings that result by elimination of $2$-torsion. + +\begin{thm} Take coefficients in a ring $R$ in which $2$ is a unit. Then +$$H^*(BO(2n)) \iso H^*(BO(2n+1))\iso H^*(BSO(2n+1)) \iso R[p_1,\ldots\!,p_n]$$ +and +$$H^*(BSO(2n))\iso R[p_1,\ldots\!,p_{n-1},e], \, \, \text{with}\, \, e^2=p_n.$$ +\end{thm} + +\section{A glimpse at the general theory} + +We should place the theory of vector bundles in a more general context. We have +written $BO(n)$, $BU(n)$, and $BSO(n)$ for certain ``classifying spaces'' in this +chapter, but we defined a classifying space $BG$ for any topological group +$G$ in Chapter 16 \S5. In fact, the spaces here are homotopy equivalent to the spaces of +the same name that we defined there, and we here explain why. + +Consider bundles $\xi: Y\rtarr B$ with fiber $G$. For spaces $U$ in a numerable open cover +$\sO$ of $B$, there are homeomorphisms $\ph: U\times G\rtarr p^{-1}(U)$ such +that $p\com \ph= \pi_1$. We say that $Y$ is a principal $G$-bundle +\index{principal G-bundle@principal $G$-bundle} if $Y$ has a free +right action by $G$, $B$ is the orbit space $Y/G$, $\xi$ is the quotient map, and the +$\ph$ are maps of right $G$-spaces. We say that $\xi: Y\rtarr B$ is a universal +principal $G$-bundle\index{universal principal G-bundle@universal principal $G$-bundle} +if $Y$ is a contractible space. In particular, for any topological group $G$ +whose identity element is a nondegenerate basepoint, such as any Lie group $G$, the map +$p: EG\rtarr BG$ constructed in Chapter 16 \S5 is a universal principal $G$-bundle. The +classification +theorem below implies that the base spaces of any two universal principal $G$-bundles are +homotopy equivalent, and it is usual to write $BG$ for any space\index{classifying space} +in this homotopy type. Observe that the long exact sequence of homotopy groups of a universal +principal $G$-bundle gives isomorphisms $\pi_q(BG)\iso \pi_{q-1}(G)$ for $q\geq 1$. + +We have implicitly constructed other examples of universal principal $G$-bundles when +$G$ is $O(n)$, $U(n)$, or $SO(n)$. To see this, consider $V_n(\bR^q)$. +Write $\bR^q=\bR^n\times \bR^{q-n}$ and note that this fixes embeddings of $O(n)$ and +$O(q-n)$ in the orthogonal group $O(q)$. Of course, $O(q)$ acts on vectors in $\bR^q$ +and thus on $n$-frames. Consider the fixed $n$-frame $x_0=\sset{e_1,\ldots\!,e_n}$. Any other +$n$-frame can be obtained from this one by the action of an element of $O(q)$, and +the isotropy group of $x_0$ is $O(q-n)$. Thus the action of $O(q)$ is transitive, and +evaluation on $x_0$ induces a homeomorphism $O(q)/O(q-n) \rtarr V_n(\bR^q)$ of $O(q)$-spaces. +The action of $O(n)\subset O(q)$ is free, and passage to orbits gives a homeomorphism +$O(q)/O(n)\times O(q-n) \rtarr G_n(\bR^q)$. It is intuitively clear and not hard to prove +that the colimit over $q$ of the inclusions $O(q-n)\rtarr O(q)$ is a homotopy equivalence +and that this implies the contractibility of $V_n(\bR^{\infty})$. We deduce that +$V_n(\bR^{\infty})$ is a universal principal $O(n)$-bundle. We have analogous universal +principal $U(n)$-bundles and $SO(n)$-bundles. + +There is a classification theorem\index{classification theorem!for principal $G$-bundles} for +principal $G$-bundles. Let $\sP G(B)$\index{PG(B)@$\sP G(B)$} denote the set +of equivalence classes of principal $G$-bundles over $B$, where two principal $G$-bundles +over $B$ are equivalent if there is a $G$-homeomorphism over $B$ between them. +Via pullback of bundles, this is a contravariant set-valued functor on the homotopy category +of spaces. + +\begin{thm} Let $\ga: Y\rtarr Y/G$ be any universal principal $G$-bundle. The natural +transformation $\PH: [-,Y/G]\rtarr \sP G(-)$ obtained by sending the +homotopy class of a map $f: B\rtarr Y/G$ to the equivalence class of the principal +$G$-bundle $f^*Y$ is a natural isomorphism of functors. +\end{thm} + +Now let $F$ be any space on which $G$ acts effectively from the left. Here an action +is effective\index{effective group action} if $gf=f$ for every $f\in F$ implies $g=e$. For a principal +$G$-bundle $Y$, let $G$ act on $Y\times F$ by $g(y,f)=(yg^{-1},gf)$ and let $Y\times_G F$ be the +orbit space $(Y\times F)/G$. With the correct formal definition of a fiber bundle +with group $G$ and fiber $F$, every such fiber bundle $p: E \rtarr B$ is equivalent to +one of the form $Y\times_G F\rtarr Y/G\iso B$ for some principal $G$-bundle $Y$ over $B$; +moreover $Y$ is uniquely determined up to equivalence. + +In fact, the ``associated principal +$G$-bundle''\index{associated principal $G$-bundle} $Y$ can be constructed +as the function space of all maps $\ps:F\rtarr E$ such that $\ps$ is an admissible +homeomorphism onto some fiber $F_b=p^{-1}(b)$. Here admissibility means that the composite +of $\ps$ with the homeomorphism $F_b\rtarr F$ determined by a coordinate chart +$\ph: U\times F\overto{\iso} p^{-1}(U)$, $b\in U$, coincides with action by some element +of $G$. The left action of $G$ on $F$ induces a right action of $G$ on $Y$; this action +is free because the given action on $F$ is effective. The projection $Y\rtarr B$ +sends $\ps$ to $b$ when $\ps: F\overto{\iso} F_b$, and it factors through a homeomorphism +$Y/G\rtarr B$. $Y$ inherits local triviality from $p$, and the evaluation map $Y\times F\rtarr E$ +induces an equivalence of bundles $Y\times_G F\rtarr E$. + +We conclude that, for any $F$, $\sP G(B)$ +is naturally isomorphic to the set of equivalence classes of bundles with group $G$ and fiber +$F$ over $B$. Fiber bundles with group $O(n)$ and fiber $\bR^n$ are real $n$-plane bundles, +fiber bundles with group $U(n)$ and fiber $\bC^n$ are complex $n$-plane bundles, and fiber +bundles with group $SO(n)$ and fiber $\bR^n$ are oriented real $n$-plane bundles. Thus the +classification theorems of the previous sections could all be rederived as special cases +of the general classification theorem for principal $G$-bundles stated in this section. + +In our discussion of Stiefel-Whitney and Chern classes, we used that passage to +classifying spaces is a product-preserving functor, at least up to homotopy. +For the functoriality, if $f: G\rtarr H$ is a homomorphism of topological groups, then +consideration of the way bundles are constructed by gluing together coordinate charts shows +that a principal $G$-bundle $\xi: Y\rtarr B$ naturally gives rise to a principal $H$-bundle +$f_*Y\rtarr B$. This construction is represented on the classifying space level by a map +$Bf: BG\rtarr BH$. + +In fact, if $EG\rtarr BG$ and $EH\rtarr BH$ are universal principal +bundles, then any map $\tilde{f}: EG\rtarr EH$ such that $\tilde{f}(xg)=\tilde{f}(x)f(g)$ +for all $x\in EG$ and $g\in G$ induces a map in the homotopy class $Bf$ on passage to orbits. +For example, if $f: G\rtarr G$ is given by conjugation by $\ga\in G$, $f(g) = \ga^{-1}g\ga$, +then $\tilde{f}(x) = x\ga$ satisfies this equivariance property and therefore $Bf$ is homotopic +to the identity. This explains why inner conjugations induce the identity map on passage to +classifying spaces, as we used in our discussion of Stiefel-Whitney and Chern classes. + +If $EG\rtarr BG$ and $EG'\rtarr BG'$ are universal principal $G$ and $G'$ bundles, then +$EG\times EG'$ is a contractible space with a free action by $G\times G'$. The orbit +space is $BG\times BG'$, and this shows that $BG\times BG'$ is a choice for the +classifying space $B(G\times G')$ and is therefore homotopy equivalent to any other choice. + +The explicit construction of $BG$ given in Chapter 16 \S5 is functorial in $G$ on the point-set +level and not just up to homotopy, and it is product preserving in the strong sense that +the projections induce a homeomorphism $B(H\times G)\iso BH\times BG$. + +\vspace{.1in} + +\begin{center} +PROBLEMS +\end{center} +\begin{enumerate} +\item Verify that $w(\bR P^q)=1$ if and only if $q=2^k-1$ for some $k$. +\item Prove that $\bR P^{2^k}$ cannot immerse in $\bR^{2^{k+1}-2}$. (By the Whitney embedding +theorem, any smooth closed $n$-manifold immerses in $\bR^{2n-1}$, so this is a best possible +non-immersion result.) +\item Prove that all tangential Stiefel-Whitney numbers of $\bR P^{q}$ are zero if and only +if $q$ is odd. +\item* Try to construct a smooth compact manifold whose boundary is $\bR P^{3}$. +\item Prove that a smooth closed $n$-manifold $M$ is $R$-orientable if and only its tangent +bundle is $R$-orientable. +\end{enumerate} + +\chapter{An introduction to $K$-theory} + +The first generalized cohomology theory to be discovered was $K$-theory, and it plays a +vital role in the connection of algebraic topology to analysis and algebraic geometry. +The fact that it is a generalized cohomology theory is a consequence of the Bott +periodicity theorem, which is one of the most important and influential theorems in all of +topology. We give some basic information about $K$-theory and, following Adams and +Atiyah, we explain how the Adams operations in $K$-theory allow a quick solution to the +``Hopf invariant one problem.'' One implication is the purely algebraic theorem that the +only possible dimensions +of a real (not necessarily associative) division algebra are 1, 2, 4, and 8. We shall only +discuss complex $K$-theory, although there is a precisely analogous construction of real +$K$-theory $KO$. From the point of view of algebraic topology, real $K$-theory is a +substantially more powerful invariant, but complex $K$-theory is usually more relevant to +applications in other fields. + +\section{The definition of $K$-theory} + +Except where otherwise noted, we work with complex vector bundles throughout this chapter. +Dimension will mean complex dimension and line bundles will mean complex line bundles. +We consider the set $Vect(X)$\index{Vect(X)@$Vect(X)$} of equivalence classes of vector bundles +over a space $X$. We assume unless otherwise specified that $X$ is compact. We remind the reader +that vector bundles can have different dimension over different components of $X$. The set +$Vect(X)$ forms an Abelian monoid (= semi-group) under Whitney sum, and it forms a semi-ring +with multiplication given by the (internal) tensor product of vector bundles over $X$. + +There is a standard construction, called the Grothendieck construction,\index{Grothendieck +construction} +of an Abelian group $G(M)$ associated to an Abelian monoid $M$: one takes the quotient of +the free Abelian group generated by the elements of $M$ by the subgroup generated by the +set of elements of the form $m+n-m\oplus n$, where $\oplus$ is the sum in $M$. The evident +morphism of Abelian monoids $i: M\rtarr G(M)$ is universal: for any homomorphism of monoids +$f: M\rtarr G$, where $G$ is an Abelian group, there is a unique homomorphism of groups +$\tilde{f}: G(M)\rtarr G$ such that $\tilde{f}\com i=f$. If $M$ is a semi-ring, then its +multiplication induces a multiplication on $G(M)$ such that $G(M)$ is a ring, called the +Grothendieck ring\index{Grothendieck ring} of $M$. If the semi-ring $M$ is commutative, then +the ring $G(M)$ is commutative. + +\begin{defn} The $K$-theory\index{K-theory@$K$-theory} of $X$, denoted $K(X)$,\index{K(X)@$K(X)$} +is the Grothendieck ring of the +semi-ring $Vect(X)$. An element of $K(X)$ is called a virtual bundle\index{virtual bundle} over $X$. +We write $[\xi]$ for the element of $K(X)$ determined by a vector bundle $\xi$. +\end{defn} + +Since $\epz$ is the identity element for the product in $K(X)$, it is standard to write +$q=[\epz^q]$, where $\epz^q$ is the $q$-dimensional trivial bundle. For vector bundles +over a based space $X$, we have the function $d: Vect(X)\rtarr \bZ$ that sends a vector +bundle to the dimension of its restriction to the component of the basepoint $*$. Since +$d$ is a homomorphism of semi-rings, it induces a dimension function\index{dimension function} +$d: K(X)\rtarr \bZ$, which is a homomorphism of rings. Since $d$ is an isomorphism +when $X$ is a point, $d$ can be identified with the induced map $K(X)\rtarr K(*)$. + +\begin{defn} The reduced $K$-theory\index{K-theory@$K$-theory!reduced} +$\tilde{K}(X)$\index{K(X)a@$\tilde K(X)$} of a based +space $X$ is the kernel +of $d: K(X)\rtarr \bZ$. It is an ideal of $K(X)$ and thus a ring without identity. +Clearly $K(X)\iso \tilde{K}(X)\times \bZ$. +\end{defn} + +We have a homotopical interpretation of these definitions, and it is for this that we +need $X$ to be compact. By the classification +theorem, we know that $\sE U_n(X)$ is naturally isomorphic to $[X_+,BU(n)]$; we have +adjoined a disjoint basepoint because we are thinking cohomologically and want the +brackets to denote based homotopy classes of maps. We have maps $i_n: BU(n)\rtarr BU(n+1)$. +With our construction of classifying spaces via Grassmannians, these maps are inclusions, +and we define $BU$ to be the colimit of the $BU(n)$, with the topology of the union. + +We say that bundles $\ze$ and $\xi$ are stably equivalent\index{stably equivalent bundles} if, +for a sufficiently large $q$, +the bundles $\ze\oplus \epz^{q-m}$ and $\xi\oplus \epz^{q-n}$ are equivalent, where +$m=d(\ze)$ and $n=d(\xi)$. Let $\sE U(X)$\index{EU(X)@$\sE U(X)$} be the set of stable +equivalence classes of vector +bundles over $X$. If $X$ is connected, or if we restrict attention to vector bundles that are +$n$-plane bundles for some $n$, then $\sE U$ is isomorphic to $\colim \sE U_n(X)$, where the +colimit is taken over the maps $\sE U_n(X)\rtarr \sE U_{n+1}(X)$ obtained by sending a bundle +$\xi$ to $\xi\oplus \epz$. Since a map from a compact space $X$ into $BU$ has image in one of the +$BU(n)$, and similarly for homotopies, we see that in this case $[X_+,BU]\iso \colim [X_+,BU(n)]$ +and therefore +$$\sE U(X)\iso [X_+,BU].$$ + +A deeper use of compactness gives the following basic fact. + +\begin{prop} If $\xi:E\rtarr X$ is a vector bundle over $X$, then there is a +bundle $\et$ over $X$ such that $\xi\oplus \et$ is equivalent to $\epz^q$ for some $q$. +\end{prop} +\begin{proof}[Sketch proof] +The space $\GA E$ of sections of $E$ is a vector space under fiberwise addition +and scalar multiplication. Using a partition of unity argument, one can show that there +is a finite dimensional vector subspace $V$ of $\GA(E)$ such that the map $g: X\times V\rtarr E$ +specified by $g(x,s)=s(x)$ is an epimorphism of bundles over $X$. The resulting short exact +sequence of vector bundles, like any other short exact sequence of vector bundles, splits as a +direct sum, and the conclusion follows. +\end{proof} + +\begin{cor} Every virtual bundle over $X$ can be written in the form $[\xi] - q$ for some +bundle $\xi$ and non-negative integer $q$. +\end{cor} +\begin{proof} +Given a virtual bundle $[\om] -[\ze]$, where $\om$ and $\ze$ are bundles, choose $\et$ such that +$\ze\oplus \et \iso \epz^q$ and let $\xi = \om\oplus \et$. Then $[\om] -[\ze] = [\xi] - q$ in $K(X)$. +\end{proof} + +\begin{cor} +There is a natural isomorphism $\sE U(X)\rtarr \tilde{K}(X)$. +\end{cor} +\begin{proof} +Writing $\sset{\xi}$ for the stable equivalence class of a bundle $\xi$, the required +isomorphism is given by the correspondence $\sset{\xi} \leftrightarrow [\xi] - d(\xi)$. +\end{proof} + +\begin{cor} Give $\bZ$ the discrete topology. For compact spaces $X$, there is a +natural isomorphism +$$K(X)\iso [X_+,BU\times \bZ].$$ +For nondegenerately based compact spaces $X$, there is a natural isomorphism +$$ \tilde{K}(X)\iso [X,BU\times \bZ].$$ +\end{cor} +\begin{proof} +When $X$ is connected, the first isomorphism sends $[\xi]-q$ to $(f,n-q)$, where $\xi$ is +an $n$-plane bundle with classifying map $f: X \rtarr BU(n)\subset BU$. The isomorphism +for non-connected spaces follows since both functors send disjoint unions to Cartesian +products. The second isomorphism follows from the first since $d: K(X)\rtarr \bZ$ can be +identified with the map $[X_+,BU\times \bZ]\rtarr [S^0,BU\times \bZ]$ induced by the +cofibration $S^0\rtarr X_+$, and the latter has kernel $[X,BU\times \bZ]$ since $X_+/S^0=X$. +\end{proof} + +For general, non-compact, spaces $X$, it is best to define $K$-theory to mean represented +$K$-theory. Here we implicitly apply CW approximation, or else use the definition in the +following form. + +\begin{defn}\index{K-theory@$K$-theory!represented} For a space $X$ of the homotopy type +of a CW complex, define +$$ K(X) = [X_+,BU\times \bZ].$$ +For a nondegenerately based space of the homotopy type of a CW complex, define +$$ \tilde{K}(X) = [X,BU\times \bZ].$$ +\end{defn} + +When $X$ is compact, we know that $K(X)$ is a ring. It is natural to expect this to remain +true for general $X$. That this is the case is a direct consequence of the following result, +which the reader should regard as an aside. + +\begin{prop} +The space $BU\times \bZ$ is a ring space\index{ring space} up to homotopy. That is, there are +additive and +multiplicative $H$-space structures on $BU\times \bZ$ such that the associativity, +commutativity, and distributivity diagrams required of a ring commute up to homotopy. +\end{prop} +\begin{proof}[Indications of proof] +By passage to colimits over $m$ and $n$, the maps $p_{m,n}: BU(m)\times BU(n) \rtarr BU(m+n)$ +induce an ``addition'' $\oplus: BU\times BU\rtarr BU$. In fact, we +can define $BU$ in terms of planes in any copy of $\bC^{\infty}$, and the explicit maps +$p_{m,n}$ of Chapter 23 \S2 pass to colimits to give +$$G_{\infty}(\bC^{\infty})\times G_{\infty}(\bC^{\infty}) +\rtarr G_{\infty}(\bC^{\infty}\oplus\bC^{\infty});$$ +use of an isomorphism $\bC^{\infty}\oplus\bC^{\infty}\iso \bC^{\infty}$ gives the required +map $\oplus$, which is well defined, associative, and commutative up to homotopy; the +zero-dimensional plane provides a convenient basepoint $0$ with which to check that we have a zero +element up to homotopy. Using ordinary addition on $\bZ$, we obtain the additive $H$-space +structure on $BU\times \bZ$. Tensor products of universal bundles give rise to classifying maps +$q_{m,n}: BU(m)\times BU(n)\rtarr BU(mn)$. These do not pass to colimits so readily, since one +must take into account the bilinearity of the tensor product, for example the relation +$(\ga_m\oplus\epz)\ten \ga_n \iso (\ga_m\ten\ga_n)\oplus \ga_n$, and we merely affirm that, by +fairly elaborate arguments, one can pass to colimits to obtain a product on $BU\times \bZ$. +It actually factors through the smash product with respect to the basepoint $0$, since that +acts as zero for the tensor product, and it restricts to an $H$-space structure on +$BO\times \sset{1}$ with basepoint $(0,1)$. +\end{proof} + +The study of ring spaces such as this is a relatively new, and quite deep, part of algebraic +topology. However, the reader should feel reasonably comfortable with the additive +$H$-space structure on $BU$. + +\section{The Bott periodicity theorem} + +There are various ways to state, and various ways to prove, this basic result. We describe +several versions and implications. One starting point is the following calculation. +We have a canonical line bundle $\ga_1^2$ over $S^2\iso \bC P^1$; its points are pairs +$(L,x)$, where $L$ is a line in $\bC^2$ and $x$ is a point on that line. We let +$H=\Hom(\ga_1^2,\epz)$ denote its dual. + +\begin{thm} $K(S^2)$ is generated as a ring by $[H]$ subject +to the single relation $([H]-1)^2=0$. Therefore, as Abelian groups, +$K(S^2)$ is free on the basis $\sset{1,[H]}$ and $\tilde{K}(S^2)$ +is free on the basis $\sset{1-[H]}$. +\end{thm} +\begin{proof}[Indication of proof] +We think of $S^2$ as the one-point compactification of $\bC$ decomposed as the +union of the unit disk $D$ and the complement $D'$ of the interior of $D$, so that +$D\cap D'=S^1$. Any $n$-plane bundle over $S^2$ restricts to a trivial bundle +over $D$ and $D'$, and these trivial bundles restrict to the same bundle over $S^1$. Conversely, +an isomorphism $f$ from the trivial bundle over $S^1$ to itself gives a way to glue together the +trivial bundles over $D$ and $D'$ to reconstruct a bundle over $S^2$. Say that two such +``clutching functions''\index{clutching function} $f$ are equivalent if the bundles +they give rise to are equivalent. +A careful analysis of the form of the possible clutching functions $f$ leads to a canonical +example in each equivalence class and thus to the required calculation. +\end{proof} + +For any pair of spaces $X$ and $Y$, we have a K\"unneth-type ring homomorphism\index{Kunneth +map@K\"unneth map} +$$\al: K(X)\ten K(Y)\rtarr K(X\times Y)$$ +specified by $\al(x\ten y) = \pi_1^*(x)\pi_2^*(y)$. + +\begin{thm}[Bott periodicity]\index{Bott periodicity} For compact spaces $X$, +$$\al: K(X)\ten K(S^2) \rtarr K(X\times S^2)$$ +is an isomorphism. +\end{thm} +\begin{proof}[Indication of proof] +The restrictions to $X\times D$ and $X\times D'$ of a bundle over $X\times S^2$ are +equivalent to pullbacks of bundles over $X$, and their further restrictions to $S^1$ +are equivalent. Conversely, bundles $\ze$ and $\xi$ over $X$ together with an +equivalence $f$ between the restrictions to $X\times S^1$ of the pullbacks of +$\ze$ and $\xi$ to $X\times D$ and $X\times D'$ determine a bundle over $X\times S^2$. +Again, a careful analysis, which is similar to that in the special case when $X=pt$, +of the equivalence classes of the possible clutching data $(\ze,f,\xi)$ +leads to the conclusion. +\end{proof} + +The following useful observation applies to any representable functor, not just $K$-theory. + +\begin{lem} For nondegenerately based spaces $X$ and $Y$, the +projections of $X\times Y$ on $X$ and on $Y$ and the quotient map $X\times Y\rtarr X\sma Y$ +induce a natural isomorphism +$$\tilde K(X\sma Y)\oplus \tilde K(X) \oplus \tilde K(Y) \iso \tilde K(X\times Y),$$ +and $\tilde K(X\sma Y)$ is the kernel of the map +$\tilde K(X\times Y)\rtarr \tilde K(X)\oplus \tilde K(Y)$ +induced by the inclusions of $X$ and $Y$ in $X\times Y$. +\end{lem} +\begin{proof} +The inclusion $X\wed Y\rtarr X\times Y$ is a cofibration with quotient $X\sma Y$, and $X$ and +$Y$ are retracts of $X\times Y$ via the inclusions and projections. +\end{proof} + +It follows easily that the K\"unneth map\index{Kunneth +map@K\"unneth map} $\al: K(X)\ten K(Y)\rtarr K(X\times Y)$ induces a +reduced K\"unneth map $\be: \tilde K(X)\ten \tilde K(Y)\rtarr \tilde K(X\sma Y)$. We have +a splitting +$$ \tilde K(X)\ten \tilde K(Y) \oplus \tilde K(X) \oplus \tilde K(Y) \oplus \bZ \iso K(X)\ten K(Y)$$ +that is compatible with the splitting of the lemma. Therefore the following reduced form of the +Bott periodicity theorem is equivalent to the unreduced form that we have already stated. + +\begin{thm}[Bott periodicity]\index{Bott periodicity} For nondegenerately based compact spaces $X$, +$$\be: \tilde K(X)\ten \tilde K(S^2) \rtarr \tilde K(X\sma S^2) = \tilde K(\SI^2 X)$$ +is an isomorphism. +\end{thm} + +Write $b=1-[H]\in\tilde K(S^2)$. Since $\tilde K(S^2)\iso \bZ$ with generator $b$, the theorem +implies that multiplication by the ``Bott element'' $b$ specifies an isomorphism +$$[X,BU\times \bZ]\iso \tilde K(X) \rtarr \tilde K(\SI^2 X)\iso [X,\OM^2(BU\times \bZ)]$$ +for nondegenerately based compact spaces $X$. Here the addition in the source and target is derived +from the natural additive $H$-space structure on $BU\times \bZ$ on the left and the displayed +double loop space on the right. If we had this isomorphism for general non-compact +spaces $X$, we could apply it with $X=BU\times \bZ$ and see that it is induced by a homotopy +equivalence of $H$-spaces +$$\be: BU\times \bZ \rtarr \OM^2(BU\times\bZ).$$ +In fact, one can deduce such a homotopy equivalence from the Bott periodicity theorem as just +stated, but there are more direct proofs. On the right, the double loop space obviously depends +only on the basepoint component $BU=BU\times\sset{0}$. Since $\pi_2(BU)=\bZ$, a little argument +with $H$-spaces shows that $\OM^2(BU\times \bZ)$ is equivalent as an $H$-space to +$(\OM^2_0 BU)\times \bZ$, where $\OM^2_0 BU$ denotes the component of the basepoint in +$\OM^2 BU$. Using the identity function on the factor $\bZ$, we see that what is +needed is an equivalence of $H$-spaces $\be: BU\rtarr \OM^2_0 BU$. In fact, it is easily deduced +from the form of Bott periodicity that, up to homotopy, $\be$ must be the adjoint of the composite +$$\xymatrix{ +\SI^2 BU = BU\sma S^2 \ar[r]^-{\id\sma b} & BU\sma BU \ar[r]^-{\ten} & BU.}$$ + +The infinite unitary group $U$ is defined to be the union of the unitary groups $U(n)$, where +$U(n)$ is embedded in $U(n+1)$ as matrices with last row and column zero except for $1$ on the +diagonal. Then $\OM BU$ is homotopy equivalent as an $H$-space to $U$. Since $\pi_1(U)=\bZ$ and +the universal cover of $U$ is the infinite special unitary group $SU$, $\OM U$ is equivalent as +an $H$-space to $(\OM SU)\times \bZ$. Therefore $\be$ may be viewed as a map +$BU\rtarr \OM SU$. Bott's original proof of the Bott periodicity theorem used the Grassmannian +model for $BU$ to write down an explicit map $\be$ in the required homotopy class and then used +Morse theory to prove that $\be$ is a homotopy equivalence. + +Bott's map $\be$ can also be proved to be a homotopy equivalence using only +basic algebraic topology. Since $BU$ and $\OM SU$ are simply connected spaces of the homotopy types +of CW complexes, a relative version of the Hurewicz theorem called the Whitehead +theorem\index{Whitehead theorem} shows that +$\be$ will be a weak equivalence and therefore a homotopy equivalence if it induces an isomorphism +on integral homology. Since $H^*(BU(n))=\bZ[c_1,\ldots\!,c_n]$, $H^*(BU)\iso \bZ[c_i|i\geq 1]$. The +$H$-space structure on $BU$ is induced by the maps $p_{m,n}$, and we find that the map +$\ps: H^*(BU)\rtarr H^*(BU\times BU)\iso H^*(BU)\ten H^*(BU)$ +induced by the product is given by $\ps(c_k)=\sum_{i+j=k}c_i\ten c_j$. A purely algebraic +dualization argument proves that, as a ring, +$$H_*(BU)\iso \bZ[\ga_i|i\geq 1],$$ +where $\ga_i$ is the image of a generator of $H_{2i}(\bC P^{\infty})$ under the map +induced by the inclusion of $\bC P^{\infty}=BU(1)$ in $BU$. One can calculate $H_*(\OM SU)$ +and see that it too is a polynomial algebra with an explicitly given generator in each +even degree. A direct inspection of the map $\be$ shows that it carries generators to generators. + +In any case, it should now be clear that we have a periodic $\OM$-prespectrum and therefore +a generalized cohomology theory represented by it. + +\begin{defn} The $K$-theory $\OM$-prespectrum $KU$\index{KU@$KU$} has spaces +$KU_{2i} = BU\times \bZ$ and +$KU_{2i+1}=U$ for all $i\geq 0$. The structure maps are given by the canonical homotopy +equivalence $U\htp \OM BU = \OM(BU\times \bZ)$ and the Bott equivalence $BU\times \bZ\htp \OM U$. +\end{defn} + +We have a resulting reduced cohomology theory\index{K-theory@$K$-theory!periodic} on based +spaces such that +$\tilde K^{2i}(X) = \tilde K(X)$ and $\tilde K^{2i+1}(X) = \tilde K(\SI X)$ +for all integers $i$. This theory has products that are induced by tensor products +of bundles over compact spaces and that are induced by suitable maps +$\ph: KU_i\sma KU_j\rtarr KU_{i+j}$ in general, just as for the cup product in +ordinary cohomology. It is standard to view this simply as a $\bZ_2$-graded +theory with groups $\tilde K^0(X)$ and $\tilde K^1(X)$. + +\section{The splitting principle and the Thom isomorphism} + +Returning to our bundle theoretic construction of $K$-theory, with $X$ compact, we describe +briefly some important generalizations of the Bott periodicity theorem. The reader should recall the +Thom isomorphism theorem in ordinary cohomology from Chapter 23 \S5. We let $\xi: E\rtarr X$ be an +$n$-plane bundle over $X$, fixed throughout this section. (We shall use the letters $E$ and $\xi$ +more or less interchangeably.) Results for general vector bundles over non-connected spaces $X$ +can be deduced by applying the results to follow to one component of $X$ at a time. + +\begin{defn} Let $E_0$ be the zero section of $E$. Define +the projective bundle\index{projective bundle} $\pi: P(E)\rtarr X$ by letting the non-zero +complex numbers +act on $E-E_0$ by scalar multiplication on fibers and taking the orbit space under this action. +Equivalently, the fiber $\pi^{-1}(x)\subset P(E)$ is the complex projective space of lines +through the origin in the fiber $\xi^{-1}(x)\subset E$. +Define the canonical line bundle $L(E)$ over $P(E)$ to be the subbundle of the pullback +$\pi^*E$ of $\xi$ along $\pi$ whose points are the pairs consisting of a line in a fiber of $E$ +and a point on that line. Let $Q(E)$ be the quotient bundle $\pi^*E/L(E)$ and let $H(E)$ denote +the dual of $L(E)$. +\end{defn} + +Observe that $P(\epz^2)=X\times \bC P^1$ is the trivial bundle over $X$ with fiber +$\bC P^1\iso S^2$. The first version of Bott periodicity generalizes, with essentially +the same proof by analysis of clutching data, to the following version. Regard $K(P(E))$ +as a $K(X)$-algebra via $\pi^*: K(X)\rtarr K(P(E))$. + +\begin{thm}[Bott periodicity]\index{Bott periodicity} Let $L$ be a line bundle over $X$ +and let $H=H(L\oplus \epz)$. Then the $K(X)$-algebra $K(P(L\oplus\epz))$ is generated by the +single element $[H]$ subject to the single relation $([H]-1)([L][H]-1)=0$. +\end{thm} + +There is a further generalization to arbitrary bundles $E$. To place it in context, we shall +first explain a cohomological analogue that expresses a different approach to the Chern classes +than the one that we sketched before. It will be based on a generalization to projective bundles +of the calculation of $H^*(\bC P^n)$. The proofs of both results are intertwined with the proof +of the following ``splitting principle,'' which allows the deduction of explicit formulas about +general bundles from formulas about sums of line bundles. + +\begin{thm}[Splitting principle]\index{splitting principle} +There is a compact space $F(E)$ and a map $p: F(E)\rtarr X$ such that $p^*E$ is a +sum of line bundles over $F(E)$ and both $p^*: H^*(X;\bZ)\rtarr H^*(F(E);\bZ)$ and +$p^*: K(X)\rtarr K(F(E))$ are monomorphisms. +\end{thm} + +This is an easy inductive consequence of the following result, which we shall refer +to as the ``splitting lemma.'' + +\begin{lem}[Splitting lemma]\index{splitting lemma} +Both $\pi^*: H^*(X;\bZ)\rtarr H^*(P(E);\bZ)$ and +$\pi^*: K(X)\rtarr K(P(E))$ are monomorphisms. +\end{lem} + +\begin{proof}[Proof of the splitting principle] +The pullback $\pi^*E$ splits as the sum $L(E)\oplus Q(E)$. (The splitting is canonically +determined by a choice of a Hermitian metric on $E$.) Applying this construction +to the bundle $Q(E)$ over $P(E)$, we obtain a map +$\pi: P(Q(E))\rtarr P(E)$ with similar properties. We obtain the desired map +$p: F(E)\rtarr X$ by so reapplying the projective bundle construction $n$ times. +Explicitly, using a Hermitian metric on $E$, we find that the fiber $F(E)_x$ is the +space of splittings of the fiber $E_x$ as a sum of $n$ lines, +and the points of the bundle $p^*E$ are $n$-tuples of vectors in given lines. +The splitting lemma implies the desired monomorphisms on cohomology and $K$-theory. +\end{proof} + +\begin{thm} +Let $x=c_1(L(E))\in H^2(P(E);\bZ)$. Then $H^*(P(E);\bZ)$ is the free $H^*(X;\bZ)$-module +on the basis $\sset{1,x,\ldots\!,x^{n-1}}$, and the Chern classes\index{Chern classes} of +$\xi$ are characterized +by $c_0(\xi)=1$ and the formula +$$\sum_{k=0}^n(-1)^kc_k(E)x^{n-k}=0.$$ +\end{thm} +\begin{proof}[Sketch proof] +This is another case where the Serre spectral sequence shows that the bundle behaves +cohomologically as if it were trivial and the K\"unneth theorem applied. This gives +the structure of $H^*(P(E))$ as an $H^*(X)$-module. In particular, it implies the +splitting lemma and thus the splitting principle in ordinary cohomology. It also implies +that there must be some description of $x^n$ as a linear combination of the $x^k$ for $k1$. The definition of $ch$ implies that the component $ch_n$ of +$ch$ in degree $2n$ is $c_n/(n-1)!$ plus terms decomposable in terms of the $c_i$ for +$i 1$. The exterior powers\index{exterior powers} of +bundles satisfy the relation +$$\la^k(\xi\oplus \et)= \oplus_{i+j=k}\la^i(\xi)\ten \la^j(\et).$$ +It follows formally that the $\la^k$ extend to operations $K(X)\rtarr K(X)$. Indeed, +form the group $G$ of power series with constant coefficient $1$ in the ring $K(X)[[t]]$ +of formal power series in the variable $t$. We define a function from (equivalence +classes of) vector bundles to this Abelian group by setting +$$\LA(\xi)=1 + \la^1(\xi)t +\cdots + \la^k(\xi)t^k +\cdots.$$ +Visibly, this is a morphism of monoids, +$$\LA(\xi\oplus\et) = \LA(\xi)\LA(\et).$$ +It therefore extend to a homomorphism of groups $\LA: K(X)\rtarr G$, and we let $\la^k(x)$ +be the coefficient of $t^k$ in $\LA(x)$. + +We define the $\ps^k$ as suitable polynomials in the $\la^k$. Recall that the subring of +symmetric polynomials in the polynomial algebra $\bZ[x_1,\ldots\!,x_n]$ is the polynomial +algebra $\bZ[\si_1,\ldots\!,\si_n]$, where $\si_i=x_1x_2\cdots x_i+\cdots$ is the +$i$th elementary symmetric function. We may write the power sum $\pi_k=x_1^k+\cdots+x_n^k$ +as a polynomial +$$\pi_k = Q_k(\si_1,\ldots\!,\si_k)$$ +in the first $k$ elementary symmetric functions. Provided $n\geq k$, $Q_k$ does not +depend on $n$. We define +$$\ps^k(x) = Q_k(\la^1(x),\ldots\!, \la^k(x)).$$ +For example, $\pi_2=\si_1^2-2\si_2$, hence $\ps^2(x)=x^2-2\la^2(x)$. The naturality of +the $\ps^k$ is clear from the naturality of the $\la^k$. + +If $\xi$ is a line bundle, then $\la^1(\xi)=\xi$ and $\la^k(\xi)=0$ for $k\geq 2$. +Clearly $\si^k_1 = \pi_k + \text{other terms}$ and $\pi_k$ does not occur as a +summand of any other monomial in the $\si_i$. Therefore $Q_k \equiv \si_1^k$ modulo +terms in the ideal generated by the $\si_i$ for $i>1$. This immediately implies +property 4. Moreover, if $\xi_1,\ldots\!,\xi_n$ are line bundles, then +\begin{eqnarray*} +\LA (\xi_1\oplus \cdots \oplus \xi_n) & = & (1+\xi_1t)\cdots(1+\xi_nt) \\ + & = & 1+\si_1(\xi_1,\ldots\!,\xi_n)t+\si_2(\xi_1,\ldots\!,\xi_n)t^2 + \cdots. +\end{eqnarray*} +This implies the generalization of property 4 to sums of line bundles: +\begin{enumerate} +\item[$4'$] $\ps^k(\xi_1\oplus\cdots\oplus \xi_n)= \pi_k(\xi_1,\ldots\!, \xi_n)$ for +line bundles $\xi_i$. +\end{enumerate} +Now, if $x$ and $y$ are sums of line bundles, the following formulas are immediate: +$$ \ps^k(x+y) = \ps^k(x)+\ps^k(y), \ \ \ps^k(xy) = \ps^k(x)\ps^k(y),\ \ +\ps^k\ps^{\ell}(x)=\ps^{k\ell}(x) $$ +$$\text{and}\ \ \ps^p(x)\equiv x^p\ \text{mod}\ p \ \ \text{for a prime}\ p. $$ +For arbitrary bundles, these formulas follow directly from the splitting principle and +naturality, and they then follow formally for arbitrary virtual bundles. This completes +the proof of all properties except 5. We have that $\tilde{K}(S^2)$ is generated by +$1-[H]$, where $(1-[H])^2=0$. +Clearly $\ps^k(1-[H]) = 1-[H]^k$. By induction on $k$, $1-[H]^k=k(1-[H])$. +Since $S^{2n}=S^2\sma\cdots\sma S^2$ and $\tilde{K}(S^{2n})$ is generated by the $k$-fold +external tensor power $(1-[H])\ten\cdots\ten(1-[H])$, property 5 follows +from the fact that $\ps^k$ preserves products. + +\begin{rem} By the splitting principle, it is clear that the $\ps^k$ are the unique +natural and additive operations with the specified behavior on line bundles. +\end{rem} + +Two further properties of the $\ps^k$ should be mentioned. The first is a direct +consequence of the multiplicativity of the $\ps^k$ and their behavior on spheres. + +\begin{prop} +The following diagram does not commute for based spaces $X$, where $\be$ is the periodicity +isomorphism: +$$\diagram +\tilde K(X)\dto_{\ps^k} \rto^(0.43){\be} & \tilde K(\SI^2 X) \dto^{\ps^k}\\ +\tilde K(X) \rto_(0.43){\be} & \tilde K(\SI^2 X).\\ +\enddiagram$$ +Rather, $\ps^k\be=k\be \ps^k$. +\end{prop} + +Therefore the $\ps^k$ do not give stable operations on the $\bZ$-graded theory $K^*$. + +\begin{prop} +Define $\ps^k_H$ on $H^{even}(X;\bZ)$ by letting $\ps^k_H(x) = k^rx$ for $x\in H^{2r}(X;\bZ)$. +Then the following diagram commutes: +$$\diagram +K(X)\dto_{\ps^k} \rto^(0.35){ch} & H^{even}(X;\bQ)\dto^{\ps^k_H}\\ +K(X) \rto_(0.35){ch} & H^{even}(X;\bQ).\\ +\enddiagram$$ +\end{prop} +\begin{proof} +It suffices to prove this on vector bundles $E$. By the splitting principle in $K$-theory +and cohomology, we may assume that $E$ is a sum of line bundles. By additivity, we may +then assume that $E$ is a line bundle. Here $\ps^k(E)= E^k$ and $c_1(E^k) = kc_1(E)$. +The conclusion follows readily from the definition of $ch$ in terms of $e^t$. +\end{proof} + +\begin{rem} The observant reader will have noticed that, by analogy with the definition +of the Stiefel-Whitney classes, we can define characteristic +classes\index{characteristic classes!in $K$-theory} in $K$-theory +by use of the Adams operations and the Thom isomorphism, setting +$\rh^k(E) = \PH^{-1}\ps^k\PH(1)$ for $n$-plane bundles $E$. +\end{rem} + +\section{The Hopf invariant one problem and its applications} + +We give one of the most beautiful and impressive illustrations of the philosophy +described in the first chapter. We define a numerical invariant, called the ``Hopf invariant,'' +of maps $f: S^{2n-1}\rtarr S^n$ and show that it can only rarely take the value one. We then +indicate several problems whose solution can be reduced to the question of when such maps +$f$ take the value one. Adams' original solution to the Hopf invariant one problem used secondary +cohomology operations in ordinary cohomology and was a critical starting point of modern algebraic +topology. The later realization that a problem that required secondary operations in ordinary +cohomology could be solved much more simply using primary operations in $K$-theory had a +profound impact on the further development of the subject. + +Take cohomology with integer coefficients unless otherwise specified. + +\begin{defn} Let $X$ be the cofiber of a based map $f: S^{2n-1}\rtarr S^n$, where $n\geq 2$. +Then $X$ is a CW complex with a single vertex, a single $n$-cell $i$, and a single $2n$-cell $j$. +The differential in the cellular chain complex of $X$ is zero for obvious dimensional +reasons, hence $\tilde H^*(X)$ is free Abelian on generators $x=[i]$ and $y=[j]$. +Define an integer $h(f)$, the Hopf invariant\index{Hopf invariant} of $f$, by $x^2 = h(f) y$. +We usually regard $h(f)$ as defined only up to sign (thus ignoring problems of orientations of +cells). Note that $h(f)$ depends only on the homotopy class of $f$. +\end{defn} + +If $n$ is odd, then $2x^2 = 0$ and thus $x^2=0$. We assume from now on that $n$ is even. +Although not essential to the main point of this section, we record the following basic +properties of the Hopf invariant. + +\begin{prop} The Hopf invariant enjoys the following properties. +\begin{enumerate} +\item If $g: S^{2n-1}\rtarr S^{2n-1}$ has degree $d$, then $h(f\com g) = dh(f)$. +\item If $e: S^n\rtarr S^n$ has degree $d$, then $h(e\com f)=d^2h(f)$. +\item The Hopf invariant defines a homomorphism $\pi_{2n-1}(S^n)\rtarr \bZ$. +\item There is a map $f: S^{2n-1}\rtarr S^n$ such that $h(f)=2$. +\end{enumerate} +\end{prop} +\begin{proof} +We leave the first three statements to the reader. For property 4, let +$\pi: D^n\rtarr D^n/S^{n-1}\iso S^n$ be the quotient map and define +$$f: S^{2n-1}\iso (D^n\times S^{n-1})\cup (S^{n-1}\times D^n)\rtarr S^n$$ +by $f(x,y)= \pi(x)$ and $f(y,x)=\pi(x)$ for $x\in D^n$ and $y\in S^{n-1}$. +We leave it to the reader to verify that $h(f)=2$. +\end{proof} + +We have adopted the standard definition of $h(f)$, but we could just as well have +defined it in terms of $K$-theory. To see this, consider the cofiber sequence +$$ S^{2n-1} \overto{f} S^n\overto{i} X \overto{\pi} S^{2n} \overto{\SI f} S^{n+1}.$$ +Obviously $i^*: H^n(X)\rtarr H^n(S^n)$ and $\pi^*: H^{2n}(S^{2n})\rtarr H^{2n}(X)$ +are isomorphisms. We have the commutative diagram with exact rows +$$\diagram +0 \rto & \tilde K(S^{2n}) \dto_{ch} \rto^{\pi^*} & \tilde K(X) \dto^{ch} \rto^{i^*} +& \tilde K(S^n) \dto^{ch}\rto & 0\\ +0 \rto & \tilde H^{**}(S^{2n};\bQ) \rto_{\pi^*} & \tilde H^{**}(X;\bQ) \rto_{i^*} +& \tilde H^{**}(S^n;\bQ) \rto & 0.\\ +\enddiagram$$ +Here the top row is exact since $\tilde K^1(S^n)=0$ and $\tilde K^1(S^{2n})=0$. The vertical +arrows are monomorphisms since they are rational isomorphisms. By a lemma in the previous +section, generators $i_n$ of $\tilde K(S^n)$ and $i_{2n}$ of $\tilde K(S^{2n})$ map under +$ch$ to generators of $H^n(S^n)$ and $H^{2n}(S^{2n})$. Choose $a\in \tilde K(X)$ such that +$i^*(a) = i_n$ and let $b=\pi^*(i_{2n})$. Then $\tilde K(X)$ is the free Abelian group on +the basis $\sset{a,b}$. Since $i_n^2=0$, we have $a^2=h'(f)b$ for some integer $h'(f)$. +The diagram implies that, up to sign, $ch(b)= y$ and $ch(a) = x+ qy$ for some rational +number $q$. Since $ch$ is a ring homomorphism and since $y^2=0$ and $xy=0$, we conclude +that $h'(f)=h(f)$. + +\begin{thm} If $h(f)=\pm 1$, then $n = 2$, $4$, or $8$. +\end{thm} +\begin{proof} +Write $n=2m$. Since $\ps^k(i_{2n})=k^{2m} i_{2n}$ and $\ps^k(i_n) = k^m i_n$, we have +$$\ps^k(b)=k^{2m} b \ \ \tand \ \ \ps^k(a) = k^m a + \mu_k b$$ +for some integer $\mu_k$. Since $\ps^2(a)\equiv a^2\ \text{mod}\ 2$, $h(f)=\pm 1$ implies +that $\mu_2$ is odd. Now, for any odd $k$, +\begin{eqnarray*} +\ps^k\ps^2(a) & = & \ps^k(2^m a + \mu_2 b) \\ +& = & k^m2^ma +(2^m\mu_k + k^{2m}\mu_2) b +\end{eqnarray*} +while +\begin{eqnarray*} +\ps^2\ps^k(a) & = & \ps^2(k^m a + \mu_k b) \\ +& = & 2^mk^ma + (k^m\mu_2 + 2^{2m} \mu_k) b. +\end{eqnarray*} +Since these must be equal, we find upon equating the coefficients of $b$ that +$$2^m(2^m-1)\mu_k = k^m(k^m-1)\mu_2.$$ +If $\mu_2$ is odd, this implies that $2^m$ divides $k^m-1$. Already with $k=3$, an +elementary number theoretic argument shows that this implies $m=1$, $2$, or $4$. +\end{proof} + +This allows us to determine which spheres can admit an $H$-space structure. Recall +from a problem in Chapter 18 that $S^{2m}$ cannot be an $H$-space. Clearly $S^n$ +is an $H$-space for $n=0$, $1$, $3$, and $7$: view $S^n$ as the unit sphere in the +space of real numbers, complex numbers, quaternions, or Cayley numbers. + +\begin{thm} +If $S^{n-1}$ is an $H$-space,\index{Hspace@$H$-space} then $n=1$, $2$, $4$, or $8$. +\end{thm} + +The strategy of proof is clear: given an $H$-space structure on $S^{n-1}$, we construct +from it a map $f: S^{2n-1}\rtarr S^n$ of Hopf invariant one. The following construction +and lemma do this and more. + +\begin{con}[Hopf construction]\index{Hopf construction} Let +$\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ be a map. +Let $CX=(X\times I)/(X\times\sset{1})$ be the unreduced cone functor and note that we have +canonical homeomorphisms of pairs +$$(D^n,S^{n-1}) \iso(CS^{n-1},S^{n-1})$$ +and +\begin{eqnarray*} +(D^{2n},S^{2n-1}) & \iso & (D^n\times D^n,(D^n\times S^{n-1})\cup (S^{n-1}\times D^n))\\ + & \iso & (CS^{n-1}\times CS^{n-1},(CS^{n-1}\times S^{n-1})\cup (S^{n-1}\times CS^{n-1})). +\end{eqnarray*} +Take $S^n$ to be the unreduced suspension of $S^{n-1}$, with the upper and lower hemispheres +$D^n_+$ and $D^n_-$ corresponding to the points with suspension coordinate $1/2\leq t\leq 1$ +and $0\leq t\leq 1/2$, respectively. Define +$$f: S^{2n-1}\iso (CS^{n-1}\times S^{n-1})\cup (S^{n-1}\times CS^{n-1}) \rtarr S^n$$ +as follows. Let $x,y\in S^{n-1}$ and $t\in I$. On $CS^{n-1}\times S^{n-1}$, $f$ is the composite +$$CS^{n-1}\times S^{n-1} \overto{\al} C(S^{n-1}\times S^{n-1}) \overto{C \ph} C S^{n-1} +\overto{\be} D^n_-,$$ +where $\al([x,t],y)=[(x,y),t]$ and $\be([x,t])=[x,(1-t)/2]$. On $S^{n-1}\times CS^{n-1}$, +$f$ is the composite +$$S^{n-1}\times CS^{n-1} \overto{\al'} C(S^{n-1}\times S^{n-1}) \overto{C \ph} C S^{n-1} +\overto{\be'} D^n_+,$$ +where $\al'(x,[y,t])=[(x,y),t]$ and $\be'([x,t])=[x,(1+t)/2]$. The map $f$, or +rather the resulting $2$-cell complex $X=S^n\cup_f D^{2n}$, is called the Hopf construction +on $\ph$. +\end{con} + +Giving $S^{n-1}$ a basepoint, we obtain inclusions of $S^{n-1}$ onto the first and second +copies of $S^{n-1}$ in $S^{n-1}\times S^{n-1}$. The bidegree\index{bidegree of a map} of a map +$\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ is the pair of integers given by the two +resulting composite maps $S^{n-1}\rtarr S^{n-1}$. Thus $\ph$ gives $S^{n-1}$ an $H$-space +structure if its bidegree is $(1,1)$. + +\begin{lem} +If the bidegree of $\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ is $(d_1,d_2)$, then the Hopf +invariant of the Hopf construction on $\ph$ is $\pm d_1d_2$. +\end{lem} +\begin{proof} +Making free use of the homeomorphisms of pairs specified in the construction, we see that the +diagonal map of $X$, its top cell $j$, evident quotient maps, and projections $\pi_i$ onto +first and second coordinates give rise to a commutative diagram in which the maps marked +$\htp$ are homotopy equivalences and those marked $\iso$ are homeomorphisms: +$$\diagram +X \rto^{\DE} \dto & X\sma X \dto^{\htp} \\ +X/S^n \rto^{\DE} & X/D^n_+ \sma X/D^n_- \\ +S^{2n}\iso D^{2n}/S^{2n-1} \drto_{\iso} \rto^(0.3){\DE} \uto_j^{\iso} +& (D^n\times D^n)/(S^{n-1}\times D^n) \sma (D^n\times D^n)/(D^n\times S^{n-1}) + \uto_{j\sma j} \dto_{\htp}^{\pi_1\sma\pi_2} \\ + & D^n/S^{n-1} \sma D^n/S^{n-1}\iso S^n\sma S^n. \\ +\enddiagram$$ +The cup square of $x\in H^n(X)$ is the image under $\DE^*$ of the external product of $x$ with +itself. The maps on the left induce isomorphisms on $H^{2n}$. The inclusions of $D^n$ +in the $i$th factor of $D^n\times D^n$ induce homotopy inverses +$$\io_1: D^n/S^{n-1}\rtarr (D^n\times D^n)/(S^{n-1}\times D^n)$$ +and +$$\io_2: D^n/S^{n-1} \rtarr (D^n\times D^n)/(D^n\times S^{n-1})$$ +to the projections $\pi_i$ in the diagram, and it suffices to prove that, up to sign, the +composites +$$j\com \io_1: D^n/S^{n-1}\rtarr X/D^n_+ \tand j\com\io_2: D^n/S^{n-1}\rtarr X/D^n_-$$ +induce multiplication by $d_1$ and by $d_2$ on $H^n$. However, by construction, these maps +factor as composites +$$D^n/S^{n-1}\overto{\ga_1}S^n/D^n_+\rtarr X/D^n_+ \tand +D^n/S^{n-1}\overto{\ga_2}S^n/D^n_-\rtarr X/D^n_-,$$ +where, up to signs and identifications of spheres, $\ga_1$ and $\ga_2$ are the suspensions +of the restrictions of $\ph$ to the two copies of $S^{n-1}$ in $S^{n-1}\times S^{n-1}$. +\end{proof} + +The determination of which spheres are $H$-spaces has the following implications. + +\begin{thm}\index{products on $\bR^n$} +Let $\om: \bR^n\times \bR^n\rtarr \bR^n$ be a map with a two-sided identity element +$e\neq 0$ and no zero divisors. Then $n=1$, $2$, $4$, or $8$. +\end{thm} +\begin{proof} +The product restricts to give $\bR^n-\sset{0}$ an $H$-space structure. Since $S^{n-1}$ +is homotopy equivalent to $\bR^n-\sset{0}$, it inherits an $H$-space structure. +Explicitly, we may assume that $e\in S^{n-1}$, by rescaling the metric, and we give +$S^{n-1}$ the product $\ph: S^{n-1}\times S^{n-1}\rtarr S^{n-1}$ specified by +$\ph(x,y)=\om(x,y)/|\om(x,y)|$. +\end{proof} + +Note that $\om$ need not be bilinear, just continuous. Also, it need not have a strict +unit; all that is required is that $e$ be a two-sided unit up to homotopy for the +restriction of $\om$ to $\bR^n-\sset{0}$. + +\begin{thm} +If $S^{n}$ is parallelizable,\index{parallelizable spheres} then $n=0$, $1$, $3$, or $7$. +\end{thm} +\begin{proof} +Exclude the trivial case $n=0$ and suppose that $S^{n}$ is parallelizable, so that its +tangent bundle $\ta$ is trivial. We will show that $S^{n}$ is an $H$-space. Define +a map $\mu: \ta \rtarr S^{n}$ as follows. Think of the tangent plane $\ta_x$ as +affinely embedded in $\bR^{n+1}$ with origin at $x$. We have a parallel translate of +this plane to an affine plane with origin at $-x$. Define $\mu$ by sending a tangent +vector $y\in \ta_x$ to the intersection with $S^{n}$ of the line from $x$ to the +translate of $y$. Composing with a trivialization $S^{n}\times\bR^n\iso \ta$, this +gives a map $\mu: S^{n}\times\bR^n\rtarr S^n$. Let $S^n_{\infty}$ be the one-point +compactification of $\bR^n$. Extend $\mu$ to a map $\ph: S^n\times S^n_{\infty}\rtarr S^n$ +by letting $\ph(x,\infty)=x$; $\ph$ is continuous since $\mu(x,y)$ approaches $x$ as +$y$ approaches $\infty$. By construction, $\infty$ is a right unit for this product. +For a fixed $x$, $y\rtarr \ph(x,y)$ is a degree one homeomorphism +$S^n_{\infty}\rtarr S^n_{\infty}$. The conclusion follows. +\end{proof} + +\chapter{An introduction to cobordism} + +Cobordism theories were introduced shortly after $K$-theory, and their use pervades +modern algebraic topology. We shall describe the cobordism of smooth closed manifolds, +but this is in fact a particularly elementary example. Other examples include smooth +closed manifolds with extra structure on their stable normal bundles: orientation, complex +structure, Spin structure, or symplectic structure for example. All of these except the +symplectic case have been computed completely. The complex case is particularly important +since complex cobordism and theories constructed from it have been of central importance in +algebraic topology for the last few decades, quite apart from their geometric origins +in the classification of manifolds. The area is pervaded by insights from algebraic +topology that are quite mysterious geometrically. For example, the complex cobordism groups +turn out to be concentrated in even degrees: every smooth closed manifold of odd dimension with +a complex structure on its stable normal bundle is the boundary of a compact manifold (with +compatible bundle information). However, there is no geometric understanding of why this +should be the case. The analogue with ``complex'' replaced by ``symplectic'' is false. + +\section{The cobordism groups of smooth closed manifolds} + +We consider the problem of classifying smooth closed $n$-manifolds $M$. One's first +thought is to try to classify them up to diffeomorphism, but that problem is in principle +unsolvable. Thom's discovery that one can classify such manifolds up to the weaker +equivalence relation of ``cobordism''\index{cobordism} is one of the most beautiful advances of +twentieth +century mathematics. We say that two smooth closed $n$-manifolds $M$ and $N$ are +cobordant\index{cobordant manifolds} +if there is a smooth compact manifold $W$ whose boundary is the disjoint union of $M$ and $N$, +$\pa W = M\amalg N$. We write $\sN_n$\index{Naa@$\sN_n$} for the set of cobordism classes of +smooth closed $n$-manifolds. It is convenient to allow the empty set $\emptyset$ as an $n$-manifold +for every $n$. Disjoint union gives an addition on the set $\sN_n$. This operation +is clearly associative and commutative and it has $\emptyset$ as a zero element. Since +$$\pa(M\times I) =M\amalg M,$$ +$M\amalg M$ is cobordant to $\emptyset$. Thus $M=-M$ and $\sN_n$ is a vector space over $\bZ_2$. +Cartesian product of manifolds defines a multiplication $\sN_m\times \sN_n\rtarr \sN_{m+n}$. +This operation is bilinear, associative, and commutative, and the zero dimensional manifold with a +single point provides an identity element. We conclude that $\sN_*$ is a graded $\bZ_2$-algebra. + +\begin{thm}[Thom]\index{Thom cobordism theorem} $\sN_*$ is a polynomial algebra over +$\bZ_2$ on generators $u_i$ of dimension +$i$ for $i > 1$ and not of the form $2^r-1$. +\end{thm} + +As already stated in our discussion of Stiefel-Whitney numbers, it follows from the proof of +the theorem that a manifold is a boundary if and only if its normal Stiefel-Whitney numbers +are zero. We can restate this as follows.\index{Stiefel-Whitney numbers} +\index{Stiefel-Whitney numbers!tangential}\index{Stiefel-Whitney numbers!normal} + +\begin{thm} Two smooth closed $n$-manifolds are cobordant if and only if their normal +Stiefel-Whitney numbers, or equivalently their tangential Stiefel-Whitney numbers, are equal. +\end{thm} + +Explicit generators $u_i$ are known. Write $[M]$ for the cobordism class of a manifold $M$. +Then we can take $u_{2i}=[\bR P^{2i}]$. We have seen that the Stiefel-Whitney numbers of +$\bR P^{2i-1}$ are zero, so we need different generators in odd dimensions. For $mq$, where $\ga_q^r$ is +the restriction of the universal bundle $\ga_q$ to the compact manifold $G_q(\bR^r)$. +By an implication of Sard's theorem known as the transversality\index{transversality} theorem, +we can deform the +restriction of $g$ to $g^{-1}(T\ga_q^r-\sset{\infty})=g^{-1}(E(\ga_q^r))$ so as to obtain a +homotopic map that is smooth and transverse to the zero section. This use of transversality +is the crux of the proof of the theorem. It follows that the inverse image +$g^{-1}(G_q(\bR^r))$ is a smooth closed $n$-manifold embedded in $\bR^{n+q}=S^{n+q}-\sset{\infty}$. +It is intuitively plausible that homotopic maps $g_i: S^{n+q}\rtarr TO(q)$, $i=0,\, 1$, give rise +to cobordant $n$-manifolds by this construction. Indeed, with the $g_i$ smooth and transverse to +the zero section, we can approximate a homotopy between them by a homotopy $h$ which is smooth on +$h^{-1}(T(\ga_q^r)-\sset{\infty})$ and transverse to the zero section. Then +$h^{-1}(G_q(\bR^{r}))$ is a manifold whose boundary +is $g_0^{-1}(G_q(\bR^r))\amalg g_1^{-1}(G_q(\bR^r))$. +It is easy to verify that the resulting function $\be: \pi_{n+q}(TO(q))\rtarr \sN_n$ is a +homomorphism. + +If we start with a manifold $M$ embedded in $\bR^{n+q}$ and construct the classifying +map $f$ for its normal bundle to be the Gauss map described in our sketch proof of +the classification theorem in Chapter 23 \S1, then the composite $Tf\com t$ is smooth and +transverse to the zero section, and the inverse image of the zero section is exactly $M$. +This proves that $\be$ is an epimorphism. To complete the proof, it suffices to show that +$\be$ is a monomorphism. It will follow formally that $\al$ is well defined and inverse +to $\be$. + +Thus suppose given $g: S^{n+q}\rtarr T\ga_q^r$ such that $g^{-1}(E(\ga_q^r))$ +is smooth and transverse to the zero section and suppose that $M=g^{-1}(G_q(\bR^r))$ is +a boundary, say $M=\pa W$. The inclusion of $M$ in $S^{n+q}$ extends to a embedding of $W$ +in $D^{n+q+1}$, by the Whitney embedding theorem for manifolds with boundary (assuming as +always that $q$ is sufficiently large). We may assume that $U=g^{-1}(T\ga_q^r-\sset{\infty})$ +is a tubular neighborhood and that $g: U \rtarr E(\ga_q^r)$ is a map of vector bundles. +A relative version of the tubular neighborhood theorem then shows that $U$ can be extended to a +tubular neighborhood $V$ of $W$ in $D^{n+q+1}$ and that $g$ extends to a map of vector bundles +$h: V\rtarr E(\ga_q^r)$. We can then extend $h$ to a map $D^{n+q+1}\rtarr T(\ga_q^r)$ by mapping +$D^{n+q+1}-V$ to $\infty$. This extension of $g$ to the disk implies that $g$ is null homotopic. + +We must still define the ring structure on $\pi_*(TO)$ and prove that we have an isomorphism +of rings and therefore of $\bZ_2$-algebras. Recall that we have maps +$p_{m,n}: BO(m)\times BO(n)\rtarr BO(m+n)$ +such that $p_{m,n}^*(\ga_{m+n})=\ga_m\times \ga_n$. The Thom space $T(\ga_m\times\ga_n)$ is +canonically homeomorphic to the smash product $TO(m)\sma TO(n)$, and the bundle map +$\ga_m\times \ga_n \rtarr \ga_{m+n}$ induces a map $\ph_{m,n}: TO(m)\sma TO(n)\rtarr TO(m+n)$. +If we have maps $f: S^{m+q}\rtarr TO(m)$ and $g: S^{n+q}\rtarr TO(n)$, then we can compose +their smash product with $\ph_{m,n}$ to obtain a composite map +$$ S^{m+n+q+r}\iso S^{m+q}\sma S^{n+r} \overto{f\sma g} TO(m)\sma TO(n) +\overto{\ph_{m,n}} TO(m+n).$$ +We can relate the maps $\ph_{m,n}$ to the maps $\si_n$. In fact, $TO$ is a commutative and +associative ring prespectrum in the sense of the following definition. + +\begin{defn} Let $T$ be a prespectrum. Then $T$ is a ring prespectrum\index{ring prespectrum} +\index{prespectrum!ring} if there are maps +$\et: S^0 \rtarr T_0$ and $\ph_{m,n}: T_m\sma T_n\rtarr T_{m+n}$ such that the +following diagrams are homotopy commutative: +$$\diagram +T_m\sma\SI T_n \ddouble \rto^{\id\sma\si_n} & T_m\sma T_{n+1} \drto^{\ph_{m,n+1}} & \\ +\SI(T_m\sma T_n) \dto_{(-1)^n} \rto^(0.56){\SI\ph_{m,n}}& \SI T_{m+n} \rto^(0.44){\si_{m+n}} & T_{m+n+1} \\ +(\SI T_m)\sma T_n \rto_(0.5){\si_m\sma \id} & T_{m+1}\sma T_n \urto_{\ph_{m+1,n}} & \\ +\enddiagram$$ + +\vspace{.1in} + +$$\diagram +S^0\sma T_n \rto^{\et\sma\id} \drto_{\iso} +& T_0\sma T_n \dto^{\ph_{0,n}} \\ +& T_n\\ +\enddiagram +\ \ \ \ \text{and} \ \ \ \ +\diagram +T_n\sma T_0 \dto_{\ph_{n,0}} & T_n\sma S^0; \lto_{\id\sma \et} \dlto^{\iso}\\ +T_{n} & \\ +\enddiagram$$ +$T$ is associative if the following diagrams are homotopy commutative: +$$\diagram +T_m\sma T_n\sma T_p \dto_{\id\sma\ph_{n,p}}\rto^(0.53){\ph_{m,n}\sma\id} +& T_{m+n}\sma T_p \dto^{\ph_{m+n,p}} \\ +T_m\sma T_{n+p} \rto_{\ph_{m,n+p}} & T_{m+n+p}; \\ +\enddiagram$$ +$T$ is commutative if there are equivalences $(-1)^{mn}: T_{m+n}\rtarr T_{m+n}$ that suspend +to $(-1)^{mn}$ on $\SI T_{m+n}$ and if the following diagrams are homotopy commutative: +$$\diagram +T_m\sma T_n \dto_{\ph_{m,n}}\rto^{t} & T_n\sma T_m \dto^{\ph_{n,m}} \\ +T_{m+n} \rto_{(-1)^{mn}} & T_{m+n}.\\ +\enddiagram$$ +When $T$ is an $\OM$-prespectrum, we can restate this as $\ph_{m,n}\htp (-1)^{mn} \ph_{n,m}t$. +\end{defn} + +For example, the Eilenberg-Mac\,Lane $\OM$-prespectrum of a commutative ring $R$ is an +associative and commutative ring prespectrum by the arguments in Chapter 22 \S3. It is +denoted $HR$ or sometimes, by abuse, $K(R,0)$. Similarly, the $K$-theory $\OM$-prespectrum +is an associative and commutative ring prespectrum. The sphere prespectrum, whose $n$th space +is $S^n$, is another example. For $TO$, the required maps $(-1)^{mn}: TO(m+n)\rtarr TO(m+n)$ +are obtained by passage to Thom complexes from a map $\ga_{m+n}\rtarr \ga_{m+n}$ of universal +bundles given on the domains of coordinate charts by the evident interchange isomorphism +$\bR^{m+n}\rtarr \bR^{m+n}$. The following lemma is immediate by passage to colimits. + +\begin{lem} +If $T$ is an associative ring prespectrum, then $\pi_*(T)$ is a graded ring. If $T$ is +commutative, then $\pi_*(T)$ is commutative in the graded sense. +\end{lem} + +Returning to the case at hand, we show that the maps $\al$ for varying $n$ transport +products of manifolds to products in $\pi_*(TO)$. Thus let $M$ be an $m$-manifold embedded in +$\bR^{m+q}$ with tubular neighborhood $U\iso E(\nu_M)$ and $N$ be an +$n$-manifold embedded in $\bR^{n+r}$ with tubular neighborhood $V\iso E(\nu_N)$. Then $M\times N$ +is embedded in $\bR^{m+q+n+r}$ with tubular neighborhood $U\times V\iso E(\nu_{M\times N})$. +Identifying +$S^{m+q+n+r}$ with $S^{m+q}\sma S^{n+r}$, we find that the Pontryagin-Thom construction +for $M\times N$ is the smash product of the Pontryagin-Thom constructions for $M$ and $N$. That is, +the left square in the following diagram commutes. The right square commutes up to homotopy by the +definition of $\ph_{q,r}$. +$$\diagram +S^{m+q}\sma S^{n+r} \rto^{t\sma t} \ddouble & T\nu_m\sma T\nu_N \dto^{\iso} \rto +& TO(q)\sma TO(r) \dto^{\ph_{q,r}} \\ +S^{m+q+n+r} \rto_{t} & T(\nu_{M\times N}) \rto & TO(q+r).\\ +\enddiagram$$ +This implies the claimed multiplicativity of the maps $\al$. + +\section{Prespectra and the algebra $H_*(TO;\bZ_2)$} + +Calculation of the homotopy groups $\pi_*(TO)$ proceeds by first computing the +homology groups $H_*(TO;\bZ_2)$ and then showing that the stable Hurewicz homomorphism +maps $\pi_*(TO)$ monomorphically onto an identifiable part of $H_*(TO;\bZ_2)$. +We explain the calculation of homology groups in this section and the next, connect the +calculation with Stiefel-Whitney numbers in \S5, and describe how to complete the +desired calculation of homotopy groups in \S6. + +We must first define the homology groups of prespectra and the stable Hurewicz homomorphism. +Just as we defined the homotopy groups of a prespectrum $T$ by the formula +$$\pi_n(T)=\colim \pi_{n+q}(T_q),$$ +we define the homology and cohomology groups\index{prespectrum!homology groups of} +\index{prespectrum!cohomology groups of} of $T$ with respect to a +homology theory $k_*$ and cohomology theory $k^*$ on spaces by the formulas +$$k_n(T)=\colim \tilde{k}_{n+q}(T_q),$$ +where the colimit is taken over the maps +$$ \tilde{k}_{n+q}(T_q) \overto{\SI_*} \tilde{k}_{n+q+1}(\SI T_q) +\overto{{\si_q}_*} \tilde{k}_{n+q+1}(T_{q+1}),$$ +and +$$k^n(T) = \lim \tilde{k}^{n+q}(T_q),$$ +where the limit is taken over the maps +$$\tilde{k}^{n+q+1}(T_{q+1})\overto{\si^*_q} \tilde{k}^{n+q+1}(\SI T_q) +\overto{\SI^{-1}} \tilde{k}^{n+q}(T_q).$$ +In fact, this definition of cohomology is inappropriate in general, differing from +the appropriate definition by a ${\lim}^1$ error term. However, the definition is +correct when $k^*$ is ordinary cohomology with coefficients in a field $R$ and each +$\tilde{H}^{n+q}(T_q;R)$ is a finite dimensional vector space over $R$. This is the +only case that we will need in the work of this chapter. In this case, it is clear +that $H^n(T;R)$ is the vector space dual of $H_n(T;R)$, a fact that we shall use +repeatedly. + +Observe that there is no cup product in $H^*(T;R)$: the maps in the +limit system factor through the reduced cohomologies of suspensions, in which +cup products are identically zero (see Problem 5 at the end of Chapter 19). +However, if $T$ is an associative and commutative ring prespectrum, then the +homology groups $H_*(T;R)$ form a graded commutative $R$-algebra. + +The Hurewicz homomorphisms $\pi_{n+q}(T_q)\rtarr \tilde{H}_{n+q}(T_q;Z)$ pass to +colimits to give the stable Hurewicz homomorphism\index{Hurewicz homomorphism!stable} +$$h: \pi_n(T)\rtarr H_n(T;\bZ).$$ +We may compose this with the map $H_n(T;\bZ)\rtarr H_n(T;R)$ induced by the unit of +a ring $R$, and we continue to denote the composite by $h$. If $T$ is an associative +and commutative ring prespectrum, then $h: \pi_*(T)\rtarr H_*(T;R)$ is a map of graded +commutative rings. + +We shall write $H_*$ and $H^*$ for homology and cohomology with coefficients in $\bZ_2$ +throughout \S\S3--6, and we tacitly assume that all homology and cohomology groups +in sight are finite dimensional $\bZ_2$-vector spaces. Recall that we have Thom isomorphisms +$$\PH_q: H^n(BO(q))\rtarr \tilde{H}^{n+q}(TO(q))$$ +obtained by cupping with the Thom class $\mu_q\in \tilde{H}^q(TO(q))$. +Naturality of the Thom diagonal applied to the map of bundles $\ga_q\oplus\epz \rtarr \ga_{q+1}$ +gives the commutative diagram +$$\diagram +\SI TO(q) \rto^(0.4){\DE} \dto_{\si_q} & BO(q)_+\sma \SI TO(q) \dto^{i_q\sma \si_q}\\ +TO(q+1) \rto_(0.35){\DE} & BO(q+1)_+\sma TO(q+1).\\ +\enddiagram$$ +This implies that the following diagram is commutative: +$$\diagram +H^n(BO(q+1))\rrto^{i_q^*} \dto_{\PH_{q+1}} & & H^n(BO(q)) \dto^{\PH_q}\\ +\tilde{H}^{n+q+1}(TO(q+1))\rto_{\si^*_q} & \tilde{H}^{n+q+1}(\SI TO(q)) +\rto_{\SI^{-1}} & \tilde{H}^{n+q}(TO(q)).\\ +\enddiagram$$ +We therefore obtain a ``stable Thom isomorphism''\index{Thom isomorphism!stable} +$$\PH: H^n(BO)\rtarr H^n(TO)$$ +on passage to limits. We have dual homology Thom isomorphisms +$$\PH_n: \tilde{H}_{n+q}(TO(q))\rtarr H_n(BO(q))$$ +that pass to colimits to give a stable Thom isomorphism +$$\PH: H_n(T) \rtarr H_n(BO).$$ + +Naturality of the Thom diagonal applied to the map of bundles $\ga_q\oplus\ga_r \rtarr \ga_{q+r}$ +gives the commutative diagram +$$\diagram +TO(q)\sma TO(r) \ddto_{\ph_{q,r}} \rto^(0.33){\DE\sma\DE} +& BO(q)_+\sma TO(q)\sma BO(r)_+ \sma TO(r) \dto^{\id\sma t\sma \id}\\ +& (BO(q)\times BO(r))_+\sma TO(q)\sma TO(r) \dto^{(p_{q,r})_+\sma \ph_{q,r}} \\ +TO(q+r) \rto_(0.38){\DE} & BO(q+r)_+\sma TO(q+r). \\ +\enddiagram$$ +As we observed for $BU$ in the previous chapter, the maps $p_{q,r}$ pass to colimits to give +$BO$ an $H$-space structure, and it follows that $H_*(BO)$ is a $\bZ_2$-algebra. On passage +to homology and colimits, these diagrams imply the following conclusion. + +\begin{prop} The Thom isomorphism $\PH: H_*(TO)\rtarr H_*(BO)$ is an isomorphism of +$\bZ_2$-algebras. +\end{prop} + +The description of the $H^*(BO(n))$ and the maps $i_q^*$ in Chapter 23 \S2 implies that +$$H^*(BO)=\bZ_2[w_i|i\geq 1]$$ +as an algebra. However, we are more interested in its ``coalgebra''\index{coalgebra} structure, +which is given by the vector space dual +$$\ps: H^*(BO)\rtarr H^*(BO)\ten H^*(BO)$$ +of its product in homology. It is clear from the description of the $p_{q,r}^*$ that +$$\ps(w_k)=\sum_{i+j=k} w_i\ten w_j.$$ +From here, determination of $H_*(BO)$ and therefore $H_*(TO)$ as an algebra is a purely algebraic, +but non-trivial, problem in dualization. Let $i: \bR P^{\infty}=BO(1)\rtarr BO$ be the inclusion. +Let $x_i\in H_i(\bR P^{\infty})$ be the unique non-zero element and let $b_i=i_*(x_i)$. +Then the solution of our dualization problem takes the following form. + +\begin{thm} $H_*(BO)$ is the polynomial algebra $\bZ_2[b_i|i\geq 1]$. +\end{thm} + +Let $a_i\in H_i(TO)$ be the element characterized by $\PH(a_i) = b_i$. + +\begin{cor} +$H_*(TO)$ is the polynomial algebra $\bZ_2[a_i|i\geq 1]$. +\end{cor} + +Using the compatibility of the Thom isomorphisms for $BO(1)$ and $BO$, we see that the +$a_i$ come from $H_*(TO(1))$. Remember that elements of $H_{i+1}(TO(1))$ map to elements +of $H_i(TO)$ in the colimit; in particular, the non-zero element of $H_1(TO(1))$ maps to +the identity element $1\in H_0(TO)$. Recall from Chapter 23 \S6 that we have a homotopy equivalence +$j: \bR P^{\infty}\rtarr TO(1)$. + +\begin{cor} For $i\geq 0$, $j_*(x_{i+1})$ maps to $a_i$ in $H_*(TO)$, where $a_0=1$. +\end{cor} + +\section{The Steenrod algebra and its coaction on $H_*(TO)$} + +Since the Steenrod operations are stable and natural, they pass to limits to define +natural operations\index{prespectrum!Steenrod operations of} +$Sq^i: H^n(T)\rtarr H^{n+i}(T)$ for $i\geq 0$ and prespectra $T$. Here +$Sq^0=\id$, but it is not true that $Sq^i(x)=0$ for $i>\deg\,x$. For example, we have the +``stable Thom class''\index{Thom class!stable} $\PH(1)=\mu\in H^0(TO)$, and it is immediate +from the definition of the +Stiefel-Whitney classes that $\PH(w_i)=Sq^i(\mu)$. Of course, $Sq^i(1)=0$ for +$i>0$, so that $\PH$ does not commute with Steenrod operations. The homology and +cohomology of $TO$ are built up from $\pi_*(TO)$ and Steenrod operations. We need +to make this statement algebraically precise to determine $\pi_*(TO)$, and we need +to assemble the Steenrod operations into an algebra to do this. + +\begin{defn} The mod $2$ Steenrod algebra\index{Steenrod algebra} $A$ is the quotient +of the free associative +$\bZ_2$-algebra generated by elements $Sq^i$, $i\geq 1$, by the ideal generated by the +Adem relations (which are stated in Chapter 22 \S5). +\end{defn} + +The following lemmas should be clear. + +\begin{lem} For spaces $X$, $H^*(X)$ has a natural $A$-module structure. +\end{lem} + +\begin{lem} For prespectra $T$, $H^*(T)$ has a natural $A$-module structure. +\end{lem} + +The elements of $A$ are stable mod $2$ cohomology operations, and our description of the cohomology +of $K(\bZ_2,q)$s in Chapter 22 \S5 implies that $A$ is in fact the algebra of all stable mod $2$ cohomology +operations, with multiplication given by composition. Passage to limits over $q$ leads to the +following lemma. Alternatively, with the more formal general definitions of the next section, +it will become yet another application of the Yoneda lemma. Recall +that $H\bZ_2$ denotes the Eilenberg-Mac\,Lane $\OM$-prespectrum $\sset{K(\bZ_2,q)}$. + +\begin{lem} As a vector space, $A$ is isomorphic to $H^*(H\bZ_2)$. +\end{lem} + +We shall see how to describe the composition in $A$ homotopically in the next section. +What is more important at the moment is that the lemma allows us to read off a basis for $A$. + +\begin{thm} $A$ has a basis consisting of the operations $Sq^I = Sq^{i_1}\cdots Sq^{i_j}$, +where $I$ runs over the sequences $\sset{i_1,\ldots\!,i_j}$ of positive integers such that +$i_{r}\geq 2 i_{r+1}$ for $1\leq r < j$. +\end{thm} + +What is still more important to us is that $A$ not only has the composition product +$A\ten A\rtarr A$, it also has a coproduct $\ps: A\rtarr A\ten A$. Giving $A\ten A$ +its natural structure as an algebra, $\ps$ is the unique map of algebras specified +on generators by $\ps(Sq^k) = \sum_{i+j=k} Sq^i\ten Sq^j$. The fact that $\ps$ is a +well defined map of algebras is a formal consequence of the Cartan formula. Algebraic +structures like this, with compatible products and coproducts, are called +``Hopf algebras.''\index{Hopf algebra} + +We write $A_*$ for the vector space dual of $A$, and we give it the dual basis to +the basis just specified on $A$. While $A_*$ is again a Hopf algebra, we are +only interested in its algebra structure at the moment. In contrast with $A$, the algebra +$A_*$ is commutative, as is apparent from the form of the coproduct on the generators of $A$. +Recall that $H\bZ_2$ is an associative and commutative ring prespectrum, so that $H_*(H\bZ_2)$ +is a commutative $\bZ_2$-algebra. The definition of the product on $H\bZ_2$ (in Chapter 22 \S3) and +the Cartan formula directly imply the following observation. + +\begin{lem} $A_*$ is isomorphic as an algebra to $H_*(H\bZ_2)$. +\end{lem} + +We need an explicit description of this algebra. In principle, this is a matter of pure +algebra from the results already stated, but the algebraic work is non-trivial. + +\begin{thm} For $r\geq 1$, define $I_r=(2^{r-1}, 2^{r-2},\ldots\!, 2, 1)$ and define $\xi_r$ +to be the basis element of $A_*$ dual to $Sq^{I_r}$. Then $A_*$ is the polynomial algebra +$\bZ_2[\xi_r|r\geq 1]$. +\end{thm} + +We need a bit of space level motivation for the particular relevance of the elements $\xi_r$. +We left the computation of the Steenrod operations in $H^*(\bR P^{\infty})$ as an exercise, +and the reader should follow up by proving the following result. + +\begin{lem} In $H^*(\bR P^{\infty})=\bZ_2[\al]$, $Sq^{I_r}(\al)=\al^{2^r}$ for $r\geq 1$ +and $Sq^{I}(\al)=0$ for all other basis elements $Sq^I$ of $A$. +\end{lem} + +The $A$-module structure maps +$$A\ten H^*(X)\rtarr H^*(X) \ \ \tand \ \ A\ten H^*(T)\rtarr H^*(T)$$ +for spaces $X$ and prespectra $T$ dualize to give ``$A_*$-comodule''\index{comodule} structure maps +$$\ga: H_*(X)\rtarr A_*\ten H_*(X) \ \ \tand \ \ \ga: H_*(T)\rtarr A_*\ten H_*(T).$$ +We remind the reader that we are implicitly assuming that all homology and cohomology groups +in sight are finitely generated $\bZ_2$-vector spaces, although these ``coactions'' can in fact be +defined without this assumption. + +Formally, the notion of a comodule $N$ over a coalgebra $C$ +is defined by reversing the direction of arrows in a diagrammatic definition of a module over +an algebra. For example, for any vector space $V$, $C\ten V$ is a comodule with action +$$\ps\ten\id: C\ten V\rtarr C\ten C\ten V.$$ +Note that, dualizing the unit of an algebra, a $\bZ_2$-coalgebra is +required to have a counit $\epz: C\rtarr \bZ_2$. We understand all of these algebraic structures +to be graded, and we say that a coalgebra is connected if $C_i=0$ +for $i<0$ and $\epz: C_0\rtarr \bZ_2$ is an isomorphism. When considering the Hurewicz homomorphism +of $\pi_*(TO)$, we shall need the following observation. + +\begin{lem} Let $C$ be a connected coalgebra and $V$ be a vector space. +An element $y\in C\ten V$ satisfies $(\ps\ten\id)(y) = 1\ten y$ if and +only if $y\in C_0\ten V\iso V$. +\end{lem} + +If $V$ is a $C$-comodule with coaction $\nu: V\rtarr C\ten V$, +then $\nu$ is a morphism of $C$-comodules. Therefore the coaction maps $\ga$ above are maps of +$A_*$-comodules for any space $X$ or prespectrum $T$. We also need the following observation, +which is implied by the Cartan formula. + +\begin{lem} If $T$ is an associative ring prespectrum, then $\ga: H_*(T)\rtarr A_*\ten H_*(T)$ +is a homomorphism of algebras. +\end{lem} + +The lemma above on Steenrod operations in $H^*(\bR P^{\infty})$ dualizes as follows. + +\begin{lem} Write the coaction $\ga: H_*(\bR P^{\infty})\rtarr A_*\ten H_*(\bR P^{\infty})$ +in the form $\ga(x_i) = \sum_j a_{i,j}\ten x_j$. Then +$$ a_{i,1}= \left\{ \begin{array}{ll} +\xi_r & \mbox{if $i=2^r$ for some $r\geq 1$}\\ +0 & \mbox{otherwise.} +\end{array} \right. $$ +\end{lem} + +Note that $a_{i,i}=1$, dualizing $Sq^0(\al^i)=\al^i$. + +Armed with this information, we return to the study of the algebra $H_*(TO)$. +We know that it is isomorphic to $H_*(BO)$, but the crux of the matter is to +redescribe it in terms of $A_*$. + +\begin{thm} Let $N_*$ be the algebra defined abstractly by +$$N_*=\bZ_2[u_i|i>1 \tand i\neq 2^r-1],$$ +where $\deg u_i = i$. Define a homomorphism of algebras $f: H_*(TO)\rtarr N_*$ by +$$ f(a_i)= \left\{ \begin{array}{ll} +u_i & \mbox{if $i$ is not of the form $2^r-1$}\\ +0 & \mbox{if $i=2^r-1$.} +\end{array} \right. $$ +Then the composite +$$g: H_*(TO)\overto{\ga} A_*\ten H_*(TO) \overto{\id\ten f} A_*\ten N_*$$ +is an isomorphism of both $A$-comodules and $\bZ_2$-algebras. +\end{thm} +\begin{proof} It is clear from things already stated that $g$ is a map of both $A$-comodules +and $\bZ_2$-algebras. We must prove that it is an isomorphism. Its source and target are both +polynomial algebras with one generator of degree $i$ for each $i\geq 1$, hence it suffices to +show that $g$ takes generators to generators. Recall that $a_i=j_*(x_{i+1})$. This allows +us to compute $\ga(a_i)$. Modulo terms that are decomposable in the algebra $A_*\ten H_*(TO)$, +we find +$$ \ga(a_i)\equiv \left\{ \begin{array}{ll} +1\ten a_i & \mbox{if $i$ is not of the form $2^r-1$}\\ +\xi_r\ten 1 + 1\ten a_{2^r-1} & \mbox{if $i=2^r-1$.} +\end{array} \right. $$ +Applying $\id\ten f$ to these elements, we obtain $1\ten u_i$ in the first case and +$\xi_r\ten 1$ in the second case. +\end{proof} + +Now consider the Hurewicz homomorphism $h: \pi_*(T)\rtarr H_*(T)$ of a prespectrum $T$. +We have the following observation, which is a direct consequence of the definition of +the Hurewicz homomorphism and the fact that $Sq^i = 0$ for $i>0$ in the cohomology of spheres. + +\begin{lem} For $x\in\pi_*(T)$, $\ga(h(x))=1\ten h(x)$. +\end{lem} + +Therefore, identifying $N_*$ as the subalgebra $\bZ_2\ten N_*$ of $A_*\ten N_*$, we see that +$g\com h$ maps $\pi_*(TO)$ to $N_*$. We shall prove the following result in \S6 +and so complete the proof of Thom's theorem. + +\begin{thm} $h: \pi_*(TO)\rtarr H_*(TO)$ is a monomorphism and $g\com h$ maps $\pi_*(TO)$ +isomorphically onto $N_*$. +\end{thm} + +\section{The relationship to Stiefel-Whitney numbers} + +We shall prove that a smooth closed $n$-manifold $M$ is a boundary if and only if all +of its normal Stiefel-Whitney numbers\index{Stiefel-Whitney numbers!normal} are zero. Polynomials +in the Stiefel-Whitney +classes are elements of $H^*(BO)$. We have seen that the normal Stiefel-Whitney numbers +of a boundary are zero, and it follows that cobordant manifolds have the same normal +Stiefel-Whitney numbers. The assignment of Stiefel-Whitney numbers to +corbordism classes of $n$-manifolds specifies a homomorphism +$$\#: H^n(BO)\ten \sN_n \rtarr \bZ_2.$$ +We claim that the following diagram is commutative: +$$\diagram +H^n(BO)\ten \sN_n \rto^(0.45){\id\ten\al} \dto_{\#} & H^n(BO)\ten \pi_n(TO) \rto^{\id\ten h} +& H^n(BO)\ten H_n(TO) \dto^{\id\ten \PH} \\ +\bZ_2 & & H^n(BO)\ten H_n(BO). \llto_{\langle \, , \, \rangle}\\ +\enddiagram$$ +To say that all normal Stiefel-Whitney numbers of $M$ are zero is to say that $w\#[M]=0$ +for all $w\in H^n(BO)$. Granted the commutativity of the diagram, this is the same as to say that +$\langle w,(\PH\com h\com \al)([M])\rangle = 0$ for all $w\in H^n(BO)$. Since +$\langle \, , \, \rangle$ is the evaluation pairing of dual vector spaces, this implies that +$(\PH\com h\com \al)([M])=0$. Since $\PH$ and $\al$ are isomorphisms and $h$ is a monomorphism, +this implies that $[M]=0$ and thus that $M$ is a boundary. + +Thus we need only prove that the diagram is commutative. Embed $M$ in $\bR^{n+q}$ with normal +bundle $\nu$ and let $f: M\rtarr BO(q)$ classify $\nu$. Then $\al([M])$ is +represented by the composite $S^{n+q}\overto{t} T\nu\overto{Tf} TO(q)$. In homology, we have +the commutative diagram +$$\diagram +\tilde{H}_{n+q}(S^{n+q}) \rto^{t_*} +& \tilde{H}_{n+q}(T\nu) \rto^{(Tf)_*} \dto^{\PH} & \tilde{H}_{n+q}(TO(q)) \dto^{\PH} \\ +& H_n(M) \rto_{f_*} & H_n(BO(q)).\\ +\enddiagram$$ +Let $i_{n+q}\in \tilde{H}_{n+q}(S^{n+q})$ be the fundamental class. By the diagram and the +definitions of $\al$ and the Hurewicz homomorphism, +$$ (f_*\com \PH\com t_*)(i_{n+q}) = +(\PH\com (Tf)_*\com t_*)(i_{n+q}) = (\PH\com h\com \al)([M]) \in H_n(BO(q)).$$ +Let $z=(\PH\com t_*)(i_{n+q})\in H_n(M)$. We claim that $z$ is the fundamental class. +Granting the claim, it follows immediately that, for $w\in H^n(BO(q))$, +\begin{eqnarray*} +w\# [M] = \langle w(\nu), z\rangle & = & \langle (f^*w(\ga_q)),(\PH\com t_*)(i_{n+q}) \rangle \\ +& = & \langle w(\ga_q), (f_*\com \PH\com t_*)(i_{n+q})\rangle \\ +& = & \langle w(\ga_q), (\PH\com h\com \al)([M])\rangle. +\end{eqnarray*} + +Thus we are reduced to proving the claim. It suffices to show +that $z$ maps to a generator of $H_n(M,M-x)$ for each $x\in M$. Since we must deal with pairs, +it is convenient to use the homeomorphism between $T\nu$ and the quotient $D(\nu)/S(\nu)$ +of the unit disk bundle by the unit sphere bundle. Recall that we have a relative cap +product +$$\cap: H^q(D(\nu),S(\nu))\ten H_{i+q}(D(\nu),S(\nu))\rtarr H_i(D(\nu)).$$ +Letting $p:D(\nu)\rtarr M$ be the projection, which of course is a homotopy equivalence, we +find that the homology Thom isomorphism +$$\PH: H_{i+q}(D(\nu),S(\nu))\rtarr H_i(M)$$ +is given by the explicit formula +$$ \PH(a) = p_*(\mu \cap a).$$ +Let $x\in U\subset M$, where $U\iso \bR^n$. Let $D(U)$ and $S(U)$ be the inverse images in +$U$ of the unit disk and unit sphere in $\bR^n$ and let $V=D(U)-S(U)$. Since $D(U)$ is +contractible, $\nu|_{D(U)}$ is trivial and thus isomorphic to $D(U)\times D^q$. Write +$$\pa(D(U)\times D^q) = (D(U)\times S^{q-1})\cup (S(U)\times D^q)$$ +and observe that we obtain a homotopy equivalence +$$t: S^{n+q} \rtarr (D(U)\times D^q)/\pa (D(U)\times D^q)\iso S^{n+q}$$ +by letting $t$ be the quotient map on the restriction of the tubular neighborhood of $\nu$ +to $D(\nu|_{D(U)})$ and letting $t$ send the complement of this restriction to the basepoint. +Interpreting $t: S^{n+q}\rtarr D(\nu)/S(\nu)$ similarly, we obtain the following commutative +diagram: +\begin{small} +$$\diagram +\tilde{H}_{n+q}(S^{n+q}) \rto^(0.3){t_*}_(0.3){\iso} \ddto_{t_*} +& H_{n+q}(D(U)\times D^q,\pa(D(U)\times D^q)) \rto^(0.63){\PH}_(0.63){\iso} \dto +& H_n(D(U),S(U)) \dto^{\iso} \\ +& H_{n+q}(D(\nu),S(\nu)\cup D(\nu|_{M-V})) +\rto^(0.6){\PH} & H_n(M,M-V) \dto^{\iso} \\ +H_{n+q}(D(\nu),S(\nu))\urto \rto_(0.55){\PH} & H_n(M) \urto \rto & H_n(M,M-x).\\ +\enddiagram$$ +\end{small} +The unlabeled arrows are induced by inclusions, and the right vertical arrows are +excision isomorphisms. The maps $\PH$ are of the general form $\PH(a)=p_*(\mu\cap a)$. +For the top map $\PH$, $\mu\in H_{n+q}(D(\nu|_{D(U)}),S(\nu|_{D(U)}))\iso H_{n+q}(S^{n+q})$, +and, up to evident isomorphisms, $\PH$ is just the inverse of the suspension isomorphism +$\tilde{H}_n(S^n) \rtarr \tilde{H}_{n+q}(S^{n+q})$. The diagram shows that $z$ maps to +a generator of $H_n(M,M-x)$, as claimed. + +\section{Spectra and the computation of $\pi_*(TO) =\pi_*(MO)$} + +We must still prove that $h:\pi_*(TO)\rtarr H_*(TO)$ is a monomorphism and +that $g\com h$ maps $\pi_*(TO)$ isomorphically onto $N_*$. Write $N$ for +the dual vector space of $N_*$. (Of course, $N$ is a coalgebra, but that +is not important for this part of our work.) Remember that the Steenrod +algebra $A$ is dual to $A_*$ and that $A\iso H^*(H\bZ_2)$. The dual of +$g:H_*(TO)\rtarr A_*\ten N_*$ is an isomorphism of $A$-modules (and of +coalgebras) $g^*: A\ten N\rtarr H^*(TO)$. Thus, if we choose a basis $\sset{y_i}$ +for $N$, where $\deg\,y_i = n_i$ say, then $H^*(TO)$ is the free graded $A$-module +on the basis $\sset{y_i}$. + +At this point, we engage in a conceptual thought exercise. We think of prespectra +as ``stable objects''\index{stable objects} that have associated homotopy, homology, and +cohomology groups. Imagine that we have a good category of stable objects, analogous to the +category of based spaces, that is equipped with all of the constructions that we +have on based spaces: wedges (= coproducts), colimits, products, limits, suspensions, +loops, homotopies, cofiber sequences, fiber sequences, smash products, function +objects, and so forth. Let us call the stable objects in our imagined category +``spectra''\index{spectrum} and call the category of such objects $\sS$.\index{S@$\sS$} +We have in mind an analogy with the notions of presheaf and sheaf. + +Whatever spectra are, there must be a way of constructing a spectrum from a +prespectrum without changing its homotopy, homology, and cohomology groups. +In turn, a based space $X$ determines the prespectrum $\SI^{\infty} X=\sset{\SI^nX}$. +The homology and cohomology groups of $\SI^{\infty} X$ are the (reduced) homology and cohomology +groups of $X$; the homotopy groups of $\SI^{\infty} X$ are the stable homotopy groups of $X$. + +Because homotopy groups, homology groups, and cohomology groups on based spaces satisfy the +weak equivalence axiom, the real domain of definition of these invariants is the category +$\bar{h}\sT$ that is obtained from the homotopy category $h\sT$ of based spaces by adjoining +inverses to the weak equivalences. This category is equivalent to the homotopy +category $h\sC$ of based CW complexes. Explicitly, the morphisms from $X$ to $Y$ in +$\bar{h}\sT$ can be defined to be the based homotopy classes of maps $\GA X\rtarr \GA Y$, +where $\GA X$ and $\GA Y$ are CW approximations of $X$ and $Y$. Composition is defined +in the evident way. + +Continuing our thought exercise, we can form the homotopy category $h\sS$ of spectra and +can define homotopy groups in terms of homotopy classes of maps from sphere spectra to +spectra. Reflection on the periodic nature of $K$-theory suggests that we should define +sphere spectra of negative dimension and define homotopy groups $\pi_q(X)$ for all integers $q$. +We say that a map of spectra is a weak equivalence if it induces an isomorphism on homotopy +groups. We can form the ``stable category''\index{stable category} $\bar{h}\sS$ from $h\sS$ +exactly as we formed the +category $\bar{h}\sT$ from $h\sT$. That is, we develop a theory of CW spectra using sphere +spectra as the domains of attaching maps. The Whitehead and cellular approximation theorems +hold, and every spectrum $X$ admits a CW approximation $\GA X\rtarr X$. We define the set +$[X,Y]$ of morphisms $X\rtarr Y$ in $\bar{h}\sS$ to be the set of homotopy classes of maps +$\GA X\rtarr \GA Y$. This is a {\em stable} category in the sense that the functor +$\SI: \bar{h}\sS \rtarr \bar{h}\sS$ is an equivalence of categories. More explicitly, the +natural maps $X\rtarr \OM\SI X$ and $\SI\OM X\rtarr X$ are isomorphisms in $\bar{h}\sS$. + +In particular, up to isomorphism, +every object in the category $\bar{h}\sS$ is a suspension, hence a double suspension. This +implies that each $[X,Y]$ is an Abelian group and composition is bilinear. Moreover, for +any map $f: X\rtarr Y$, the canonical map $Ff\rtarr \OM Cf$ and its adjoint $\SI Ff\rtarr Cf$ +(see Chapter 8 \S7) are also isomorphisms in $\bar{h}\sS$, so that cofiber sequences and +fiber sequences are equivalent. Therefore cofiber sequences give rise to long exact sequences +of homotopy groups. + +The homotopy groups of wedges and products of spectra are given by +$$\pi_*(\textstyle{\bigvee}_i\, X_i) = \textstyle{\sum}_i\, \pi_*(X_i) +\tand \pi_*(\textstyle{\prod}_i\, X_i)=\textstyle{\prod}_i\, \pi_*(X_i).$$ +Therefore, if only finitely many $\pi_q(X_i)$ are non-zero for each $q$, then the natural +map $\bigvee_i\, X_i\rtarr \prod_i\, X_i$ is an isomorphism. + +We have homology groups and cohomology groups defined on $\bar{h}\sS$. A spectrum $E$ +represents a homology theory\index{homology theory!on spectra} $E_*$ and a cohomology +theory\index{cohomology theory!on spectra} $E^*$ specified in terms +of smash products and function spectra by +$$E_q(X) =\pi_q(X\sma E) \ \tand \ E^q(X) = \pi_{-q}F(X,E) \iso [X,\SI^qE].$$ +Verifications of the exactness, suspension, additivity, and weak equivalence axioms are +immediate from the properties of the category $\bar{h}\sS$. Moreover, +every homology or cohomology theory on $\bar{h}\sS$ is so represented by some spectrum $E$. + +As will become clear later, $\OM$-prespectra are more like spectra than general prespectra, +and we continue to write $H\pi$ for the ``Eilenberg-Mac\,Lane spectrum'' +\index{Eilenberg-Mac\,Lane spectrum} that represents +ordinary cohomology with coefficients in $\pi$. Its only non-zero homotopy group is +$\pi_0(H\pi)=\pi$, and the Hurewicz homomorphism maps this group isomorphically onto +$H_0(H\pi;\bZ)$. When $\pi=\bZ_2$, the natural map $H_0(H\bZ_2;\bZ)\rtarr H_0(H\bZ_2;\bZ_2)$ +is also an isomorphism. + +Returning to our motivating example, we write $MO$\index{MO@$MO$} for the +``Thom spectrum''\index{Thom spectrum} +that arises from the Thom prespectrum $TO$. The reader may sympathize +with a student who claimed that +$MO$ stands for ``Mythical Object.''\index{Mythical Object} + +We may choose a map $\bar{y}_i: MO \rtarr \SI^{n_i}H\bZ_2$ that represents the +element $y_i$. Define $K(N_*)$ to be the wedge of a copy of $\SI^{n_i}H\bZ_2$ for +each basis element $y_i$ and note that $K(N_*)$ is isomorphic in $\bar{h}\sS$ to +the product of a copy of $\SI^{n_i}H\bZ_2$ for each $y_i$. We think of $K(N_*)$ as a +``generalized Eilenberg-Mac\,Lane spectrum.'' It satisfies $\pi_*(K(N_*))\iso N_*$ +(as Abelian groups and so as $\bZ_2$-vector spaces), and the mod $2$ Hurewicz +homomorphism $h: \pi_*(K(N_*))\rtarr H_*(K(N_*))$ is a monomorphism. Using the +$\bar{y}_i$ as coordinates, we obtain a map +$$\om: MO\rtarr \textstyle{\prod}_i\, \SI^{n_i} H\bZ_2 \htp K(N_*).$$ + +The induced map $\om^*$ on mod $2$ cohomology is an isomorphism of $A$-modules: $H^*(MO)$ +and $H^*(K(N_*))$ are free $A$-modules, and we have defined $\om$ +so that $\om^*$ sends basis elements to basis elements. Therefore the induced map on +homology groups is an isomorphism. Here we are using mod $2$ homology, but it +can be deduced from the fact that both $\pi_*(MO)$ and $\pi_*(K(N_*))$ +are $\bZ_2$-vector spaces that $\om$ induces an isomorphism on integral homology groups. +Therefore the integral homology groups of $C\om$ are zero. By the Hurewicz theorem in +$\bar{h}\sS$, the homotopy groups of $C\om$ are also zero. Therefore $\om$ induces +an isomorphism of homotopy groups. That is, $\om$ is an isomorphism in $\bar{h}\sS$. +Therefore $\pi_*(MO)\iso N_*$ and the Hurewicz homomorphism $h:\pi_*(MO)\rtarr H_*(MO)$ +is a monomorphism. It follows that $g\com h:\pi_*(MO)\rtarr N_*$ is an isomorphism since +it is a monomorphism between vector spaces of the same finite dimension in each degree. + +\section{An introduction to the stable category} + +To give content to the argument just sketched, we should construct a good category of spectra. +In fact, no such category was available when Thom first proved his theorem in 1960. With +motivation from the introduction of $K$-theory and cobordism, a good stable category was +constructed by Boardman (unpublished) around 1964 and an exposition of his category was +given by Adams soon after. However, these early constructions were far more primitive than +our outline suggests. While they gave a satisfactory stable category, the underlying category +of spectra did not have products, limits, and function objects, and its smash product was +not associative, commutative, or unital. In fact, a fully satisfactory category of spectra +was not constructed until 1995. + +We give a few definitions to indicate what is involved. + +\begin{defn} +A spectrum\index{spectrum} $E$ is a prespectrum +such that the adjoints $\tilde{\si}: E_n\rtarr \OM E_{n+1}$ of the structure maps +$\si: \SI E_n \rtarr E_{n+1}$ are {\em homeomorphisms}. A map $f: T\rtarr T'$ of prespectra +is a sequence of maps $f_n: T_n\rtarr T'_n$ such that $\si'_n\com \SI f_n = f_{n+1}\com \si_n$ +for all $n$. A map $f:E\rtarr E'$ of spectra is a map between $E$ and $E'$ regarded as +prespectra. +\end{defn} + +We have a forgetful functor from the category $\sS$\index{S@$\sS$} of spectra to the +category $\sP$\index{P@$\sP$} of +prespectra. It has a left adjoint $L:\sP\rtarr \sS$. In $\sP$, we define wedges, colimits, +products, and limits spacewise. For example, $(T\wed T')_n = T_n\wed T'_n$, with the +evident structure maps. We define wedges and colimits of spectra by first performing the +construction on the prespectrum level and then applying the functor $L$. If we start with +spectra and construct products or limits spacewise, then the result is again a spectrum; +that is, limits of spectra are the limits of their underlying prespectra. Thus the category +$\sS$ is complete and cocomplete. + +Similarly, we define the smash product $T\sma X$ and function prespectrum $F(X,T)$ of a +based space $X$ and a prespectrum $T$ spacewise. For a spectrum $E$, we define $E\sma X$ +by applying $L$ to the prespectrum level construction; the prespectrum $F(X,E)$ is already +a spectrum. We now have cylinders $E\sma I_+$ and thus can define homotopies between maps +of spectra. Similarly we have cones $CE=E\sma I$ (where $I$ has basepoint $1$), suspensions +$\SI E=E\sma S^1$, path spectra $F(I,E)$ (where $I$ has basepoint $0$), and loop spectra +$\OM E =F(S^1,E)$. The development of cofiber and fiber sequences proceeds exactly as for +based spaces. + +The left adjoint $L$ can easily be described explicitly on those prespectra $T$ whose +adjoint structure maps $\tilde{\si}_n: T_n\rtarr \OM T_{n+1}$ are inclusions: +we define $(LT)_n$ to be the union of the expanding sequence +$$T_n \overto{\tilde{\si}_n} \OM T_{n+1} \overto{\OM\tilde{\si}_{n+1}} \OM^2 T_{n+2} \rtarr \cdots.$$ +We then have +$$\OM (LT)_{n+1} =\OM(\bigcup \OM^qT_{n+1+q}) \iso \bigcup \OM^{q+1}T_{n+q+1} \iso (LT)_n.$$ + +We have an evident map of prespectra $\la: T\rtarr LT$, and a comparison of colimits shows +(by a cofinality argument) that $\la$ induces isomorphisms on homotopy and homology groups. +The essential point is that homotopy and homology commute with colimits. It is not true +that cohomology converts colimits to limits in general, because of ${\lim}^1$ error terms, and +this is one reason that our definition of the cohomology of prespectra via limits is +inappropriate except under restrictions that guarantee the vanishing of ${\lim}^1$ terms. +Observe that there is no problem in the case of $\OM$-prespectra, for which $\la$ is a +spacewise weak equivalence. + +For a based space $X$, we define the suspension spectrum\index{suspension spectrum} +$\SI^{\infty}X$ by applying $L$ to the suspension prespectrum $\SI^{\infty} X =\sset{\SI^nX}$. +The inclusion condition is satisfied in this case. We define $QX=\cup \OM^q\SI^q X$,\index{QX@$QX$} +and we find that the $n$th space of $\SI^{\infty} X$ is $Q\SI^n X$. It should be apparent that +the homotopy groups of the space $QX$ are the stable homotopy groups of $X$. + +The adjoint structure maps of the Thom prespectrum $TO$ are also inclusions, and our mythical +object is $MO=L TO$.\index{MO@$MO$} + +In general, for a prespectrum $T$, we can apply an iterated mapping cylinder construction to +define a spacewise equivalent prespectrum $KT$ whose adjoint structure maps are inclusions. +The prespectrum level homotopy, homology, and cohomology groups of $KT$ are isomorphic to +those of $T$. Thus, if we have a prespectrum $T$ whose invariants we are interested in, such as +an Eilenberg-Mac\,Lane $\OM$-prespectrum or the $K$-theory $\OM$-prespectrum, then we can +construct a spectrum $LKT$ that has the same invariants. + +For a based space $X$ and $q\geq 0$, we construct a prespectrum $\SI^{\infty}_qX$ whose +$n$th space is a point for $n0$. The definition is appropriate since +$\SI S^q \iso S^{q+1}$ for all integers $q$. We can now define homotopy groups in the +obvious way. For example, the homotopy groups of the $K$-theory spectrum are $\bZ$ for every +even integer and zero for every odd integer. + +From here, we can go on to define CW spectra in very much the same way that we defined +CW complexes, and we can fill in the rest of the outline in the previous section. The real +work involves the smash product of spectra, but this does not belong in our rapid course. +While there is a good deal of foundational work involved, there is also considerable payoff +in explicit concrete calculations, as the computation of $\pi_*(MO)$ well illustrates. + +With the hope that this glimpse into the world of stable homotopy theory has whetted the +reader's appetite for more, we will end at this starting point. + +\clearpage + +\thispagestyle{empty} + +\chapter*{Suggestions for further reading} + +\setcounter{section}{0} + +Rather than attempt a complete bibliography, I will give a number of basic references. +I will begin with historical references and textbooks. I will then give references for +specific topics, more or less in the order in which topics appear in the text. Where +material has been collected in one or another book, I have often referred to such books +rather than to original articles. However, the importance and quality of exposition of +some of the original sources often make them still to be preferred today. The subject in +its earlier days was blessed with some of the finest expositors of mathematics, for example +Steenrod, Serre, Milnor, and Adams. Some of the references are intended to give historical +perspective, some are classical papers in the subject, some are follow-ups to material in +the text, and some give an idea of the current state of the subject. In fact, +many major parts of algebraic topology are nowhere mentioned in any of the existing +textbooks, although several were well established by the mid-1970s. I will indicate +particularly accessible references for some of them; the reader can find more of the +original references in the sources given. + +\section{A classic book and historical references} + +The axioms for homology and cohomology theories were set out in the classic: + +\vspace{1mm} + +\noindent +{\em S. Eilenberg and N. Steenrod. Foundations of algebraic topology.} +Princeton University Press. 1952. + +\vspace{1.3mm} + +I believe the only historical monograph on the subject is: + +\vspace{1mm} + +\noindent +{\em J. Dieudonn\'e. A history of algebraic and differential topology, 1900--1960.} +Birk\-h\"auser. 1989. + +\vspace{1.3mm} + +A large collection of historical essays will appear soon: + +\vspace{1mm} + +\noindent +{\em I.M. James, editor. The history of topology.} Elsevier Science. To appear. + +\vspace{1.3mm} + +Among the contributions, I will advertise one of my own, available on the web: + +\vspace{1mm} + +\noindent +{\em J.P. May. Stable algebraic topology, 1945--1966.} http://hopf.math.purdue.edu + +\section{Textbooks in algebraic topology and homotopy theory} + +These are ordered roughly chronologically (although this is obscured by the fact that +the most recent editions or versions are cited). I have included only those texts that +I have looked at myself, that are at least at the level of the more elementary chapters +here, and that offer significant individuality of treatment. There are many other textbooks in +algebraic topology. + +\vspace{1mm} + +Two classic early textbooks: + +\vspace{1.3mm} + +\noindent +{\em P.J. Hilton and S. Wylie. Homology theory.} Cambridge University Press. 1960. + +\vspace{1mm} + +\noindent +{\em E. Spanier. Algebraic topology.} McGraw-Hill. 1966. + +\vspace{1.3mm} + +An idiosyncratic pre-homology level book giving much material about groupoids: + +\vspace{1mm} + +\noindent +{\em R. Brown. Topology. A geometric account of general topology, homotopy types, +and the fundamental groupoid.} Second edition. Ellis Horwood. 1988. + +\vspace{1.3mm} + +A homotopical introduction close to the spirit of this book: + +\vspace{1mm} + +\noindent +{\em B. Gray. Homotopy theory, an introduction to algebraic topology.} Academic Press. 1975. + +\vspace{1.3mm} + +The standard current textbooks in basic algebraic topology: + +\vspace{1mm} + +\noindent +{\em M.J. Greenberg and J. R. Harper. Algebraic topology, a first course.} +Benjamin/\linebreak +Cummings. 1981. + +\vspace{1mm} + +\noindent +{\em W.S. Massey. A basic course in algebraic topology.} Springer-Verlag. 1991. + +\vspace{1mm} + +\noindent +{\em A. Dold. Lectures on algebraic topology.} Reprint of the 1972 edition. +Springer-Verlag. 1995. + +\vspace{1mm} + +\noindent +{\em J.W. Vick. Homology theory; an introduction to algebraic topology.} +Second edition. Springer-Verlag. 1994. + +\vspace{1mm} + +\noindent +{\em J.R. Munkres. Elements of algebraic topology.} Addison Wesley. 1984. + +\vspace{1mm} + +\noindent +{\em J.J. Rotman. An introduction to algebraic topology.} Springer-Verlag. 1986. + +\vspace{1mm} + +\noindent +{\em G.E. Bredon. Topology and geometry.} Springer-Verlag. 1993. + +\vspace{1.3mm} + +Sadly, the following are still the only more advanced textbooks in the subject: + +\vspace{1mm} + +\noindent +{\em R.M. Switzer. Algebraic topology. Homotopy and homology.} Springer-Verlag. 1975. + +\vspace{1mm} + +\noindent +{\em G.\!W. Whitehead. Elements of homotopy theory.} Springer-Verlag. 1978. + +\section{Books on CW complexes} + +Two books giving more detailed studies of CW complexes than are found in textbooks +(the second giving a little of the theory of compactly generated spaces): + +\vspace{1mm} + +\noindent +{\em A.T. Lundell and S. Weingram The topology of CW complexes.} +Van Nostrand Reinhold. 1969. + +\vspace{1mm} + +\noindent +{\em R. Fritsch and R.A. Piccinini. Cellular structures in topology.} +Cambridge University Press. 1990. + +\section{Differential forms and Morse theory} + +Two introductions to algebraic topology starting from de Rham cohomology: + +\vspace{1mm} + +\noindent +{\em R. Bott and L.\!W. Tu. Differential forms in algebraic topology.} Springer-Verlag. 1982. + +\vspace{1mm} + +\noindent +{\em I. Madsen and J. Tornehave. From calculus to cohomology. de Rham cohomology and +characteristic classes.} Cambridge University Press. 1997. + +\vspace{1.3mm} + +The classic reference on Morse theory, with an exposition of the Bott periodicity theorem: + +\vspace{1mm} + +\noindent +{\em J. Milnor. Morse theory.} Annals of Math. Studies No. 51. Princeton University Press. 1963. + +\vspace{1.3mm} + +A modern use of Morse theory for the analytic construction of homology: + +\vspace{1mm} + +\noindent +{\em M. Schwarz. Morse homology.} Progress in Math. Vol. 111. Birkh\"auser. 1993. + +%R. Bott. An application of the Morse theory to the topology of Lie-groups. +%Bull. Soc. Math. France 84(1956), 251-281. +%R. Bott. The stable homotopy of the classical groups. Annals of Math. 70(1959), 313-337. +%R. Bott. Quelques remarques sur les th\'eor\`emes de periodicit\'e de topology. Bull. Soc. +%Math. France 87(1959), 293-310. +%M.F. Atiyah and R. Bott. On the periodicity theorem for complex vector bundles. +%Acta Math. 112(1964), 229-247. + +\section{Equivariant algebraic topology} + +Two good basic references on equivariant algebraic topology, classically +called the theory of transformation groups (see also \S\S16, 21 below): + +\vspace{1mm} + +\noindent +{\em G. Bredon. Introduction to compact transformation groups.} Academic Press. 1972. + +\vspace{1mm} + +\noindent +{\em T. tom Dieck. Transformation groups.} Walter de Gruyter. 1987. + +\vspace{1.3mm} + +A more advanced book, a precursor to much recent work in the area: + +\vspace{1mm} + +\noindent +{\em T. tom Dieck. Transformation groups and representation theory.} +Lecture Notes in Mathematics Vol. 766. Springer-Verlag. 1979. + +\section{Category theory and homological algebra} + +A revision of the following classic on basic category theory is in preparation: + +\vspace{1mm} + +\noindent +{\em S. Mac\,Lane. Categories for the working mathematician.} Springer-Verlag. 1971. + +\vspace{1.3mm} + +Two classical treatments and a good modern treatment of homological algebra: + +\vspace{1mm} + +\noindent +{\em H. Cartan and S. Eilenberg. Homological algebra.} Princeton University Press. 1956. + +\vspace{1mm} + +\noindent +{\em S. MacLane. Homology.} Springer-Verlag. 1963. + +\vspace{1mm} + +\noindent +{\em C.A. Weibel. An introduction to homological algebra.} Cambridge University Press. 1994. + +\section{Simplicial sets in algebraic topology} + +Two older treatments and a comprehensive modern treatment: + +\vspace{1mm} + +\noindent +{\em P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory.} Springer-Verlag. 1967. + +\vspace{1mm} + +\noindent +{\em J.P. May. Simplicial objects in algebraic topology.} D. Van Nostrand 1967; +reprinted by the University of Chicago Press 1982 and 1992. + +\vspace{1mm} + +\noindent +{\em P.G. Goerss and J.F. Jardine. Simplicial homotopy theory.} Birkh\"auser. To appear. + +\section{The Serre spectral sequence and Serre class theory} + +Two classic papers of Serre: + +\vspace{1mm} + +\noindent +{\em J.-P. Serre. Homologie singuli\'ere des espaces fibr\'es. Applications.} Annals +of Math. (2)54(1951), 425--505. + +\vspace{1mm} + +\noindent +{\em J.-P. Serre. Groupes d'homotopie et classes de groupes ab\'eliens.} Annals of +Math. (2)58(1953), 198--232. + +\vspace{1.3mm} + +A nice exposition of some basic homotopy theory and of Serre's work: + +\vspace{1mm} + +\noindent +{\em S.-T. Hu. Homotopy theory.} Academic Press. 1959. + +\vspace{1.3mm} + +Many of the textbooks cited in \S2 also treat the Serre spectral sequence. + +\section{The Eilenberg-Moore spectral sequence} + +There are other important spectral sequences in the context of fibrations, +mainly due to Eilenberg and Moore. Three references: + +\vspace{1mm} + +\noindent +{\em S. Eilenberg and J.C. Moore. Homology and fibrations, I.} Comm. Math. Helv. +40(1966), 199--236. + +\vspace{1mm} + +\noindent +{\em L. Smith. Homological algebra and the Eilenberg-Moore spectral sequences.} +Trans. Amer. Math. Soc. 129(1967), 58--93. + +\vspace{1mm} + +\noindent +{\em V.K.A.M. Gugenheim and J.P. May. On the theory and applications of differential +torsion products.} Memoirs Amer. Math. Soc. No. 142. 1974. + +\vspace{1.3mm} + +There is a useful guidebook to spectral sequences: + +\vspace{1mm} + +\noindent +{\em J. McCleary. User's guide to spectral sequences.} Publish or Perish. 1985. + +\section{Cohomology operations} + +A compendium of the work of Steenrod and others on the construction and analysis +of the Steenrod operations: + +\vspace{1mm} + +\noindent +{\em N.E. Steenrod and D.B.A. Epstein. Cohomology operations.} Annals of Math. Studies No. 50. +Princeton University Press. 1962. + +\vspace{1.3mm} + +A classic paper that first formalized cohomology operations, among other things: + +\vspace{1mm} + +\noindent +{\em J.-P. Serre. Cohomologie modulo $2$ des complexes d'Eilenberg-Mac\,Lane.} +Comm. Math. Helv. 27(1953), 198--232. + +\vspace{1.3mm} + +A general treatment of Steenrod-like operations: + +\vspace{1mm} + +\noindent +{\em J.P. May. A general algebraic approach to Steenrod operations.} In Lecture Notes +in Mathematics Vol. 168, 153--231. Springer-Verlag. 1970. + +\vspace{1.3mm} + +A nice book on mod $2$ Steenrod operations and the Adams spectral sequence: + +\vspace{1mm} + +\noindent +{\em R. Mosher and M. Tangora. Cohomology operations and applications in homotopy theory.} +Harper and Row. 1968. + +\section{Vector bundles} + +A classic and a more recent standard treatment that includes $K$-theory: + +\vspace{1mm} + +\noindent +{\em N.E. Steenrod. Topology of fibre bundles.} Princeton University Press. +1951. Fifth printing, 1965. + +\vspace{1mm} + +\noindent +{\em D. Husemoller. Fibre bundles.} Springer-Verlag. 1966. Third edition, 1994. + +\vspace{1.3mm} + +A general treatment of classification theorems for bundles and fibrations: + +\vspace{1mm} + +\noindent +{\em J.P. May. Classifying spaces and fibrations.} Memoirs Amer. Math. Soc. No. 155. 1975. + +\section{Characteristic classes} + +The classic introduction to characteristic classes: + +\vspace{1mm} + +\noindent +{\em J. Milnor and J.D. Stasheff. Characteristic classes.} Annals of Math. Studies No. 76. +Princeton University Press. 1974. + +\vspace{1.3mm} + +A good reference for the basic calculations of characteristic classes: + +\vspace{1mm} + +\noindent +{\em A. Borel. Topology of Lie groups and characteristic classes.} Bull. Amer. Math. Soc. +61(1955), 297--432. + +\vspace{1.3mm} + +Two proofs of the Bott periodicity theorem that only use standard techniques of algebraic +topology, starting from characteristic class calculations: + +\vspace{1mm} + +\noindent +{\em H. Cartan et al. P\'eriodicit\'e des groupes d'homotopie stables des groupes +classiques, d'apr\`es Bott.} S\'eminaire Henri Cartan, 1959/60. Ecole Normale Sup\'erieure. Paris. + +\vspace{1mm} + +\noindent +{\em E. Dyer and R.K. Lashof. A topological proof of the Bott periodicity theorems.} +Ann. Mat. Pure Appl. (4)54(1961), 231--254. + +\section{$K$-theory} + +%M.F. Atiyah and F. Hirzebruch. Vector bundles and homogeneous spaces, in Differential +%Geometry. Amer. Math. Soc. Proc. Symp. Pure Math 3(1961), 7--38. + +Two classical lecture notes on $K$-theory: + +\vspace{1mm} + +\noindent +{\em R. Bott. Lectures on $K(X)$.} W.A. Benjamin. 1969. + +\vspace{1mm} + +This includes a reprint of perhaps the most accessible proof of the complex +case of the Bott periodicity theorem, namely: + +\vspace{1mm} + +\noindent +{\em M.F. Atiyah and R. Bott. On the periodicity theorem for complex vector bundles.} +Acta Math. 112(1994), 229--247. + +\vspace{1.3mm} + +\noindent +{\em M.F. Atiyah. $K$-theory.} Notes by D.W. Anderson. Second Edition. +Addison-Wesley. 1967. + +\vspace{1mm} + +This includes reprints of two classic papers of Atiyah, one that relates Adams +operations in $K$-theory to Steenrod operations in cohomology and another that +sheds insight on the relationship between real and complex $K$-theory: + +\vspace{1mm} + +\noindent +{\em M.F. Atiyah. Power operations in $K$-theory.} Quart. J. Math. (Oxford) (2)17(1966), +165--193. + +\vspace{1mm} + +\noindent +{\em M.F. Atiyah. $K$-theory and reality.} Quart. J. Math. (Oxford) (2)17(1966), 367--386. + +\vspace{1.3mm} + +Another classic paper that greatly illuminates real $K$-theory: + +\vspace{1mm} + +\noindent +{\em M.F. Atiyah, R. Bott, and A. Shapiro. Clifford algebras.} Topology +3(1964), suppl. 1, 3--38. + +\vspace{1.3mm} + +A more recent book on $K$-theory: + +\noindent +{\em M. Karoubi. $K$-theory.} Springer-Verlag. 1978. + +\vspace{1.3mm} + +Some basic papers of Adams and Adams and Atiyah giving applications of $K$-theory: + +\vspace{1mm} + +\noindent +{\em J.F. Adams. Vector fields on spheres.} Annals of Math. 75(1962), 603--632. + +\vspace{1mm} + +\noindent +{\em J.F. Adams. On the groups $J(X)$ I, II, III, and IV.} Topology 2(1963), 181--195; +3(1965), 137-171 and 193--222; 5(1966), 21--71. + +\vspace{1mm} + +\noindent +{\em J.F. Adams and M.F. Atiyah. $K$-theory and the Hopf invariant.} Quart. J. Math. (Oxford) +(2)17(1966), 31--38. + +\section{Hopf algebras; the Steenrod algebra, Adams spectral sequence} + +The basic source for the structure theory of (connected) Hopf algebras: + +\vspace{1mm} + +\noindent +{\em J. Milnor and J.C. Moore. On the structure of Hopf algebras.} Annals of Math. 81(1965), +211--264. + +\vspace{1.3mm} + +The classic analysis of the structure of the Steenrod algebra as a Hopf algebra: + +\vspace{1mm} + +\noindent +{\em J. Milnor. The Steenrod algebra and its dual.} Annals of Math. 67(1958), 150--171. + +\vspace{1.3mm} + +Two classic papers of Adams; the first constructs the Adams spectral sequence +relating the Steenrod algebra to stable homotopy groups and the second uses +secondary cohomology operations to solve the Hopf invariant one problem: + +\vspace{1mm} + +\noindent +{\em J.F. Adams. On the structure and applications of the Steenrod algebra.} +Comm. Math. Helv. 32(1958), 180--214. + +\vspace{1mm} + +\noindent +{\em J.F. Adams. On the non-existence of elements of Hopf invariant one.} +Annals of Math. 72(1960), 20--104. + +\section{Cobordism} + +The beautiful classic paper of Thom is still highly recommended: + +\vspace{1mm} + +\noindent +{\em R. Thom. Quelques propri\'et\'es globals des vari\'et\'es diff\'erentiables.} +Comm. Math. Helv. 28(1954), 17--86. + +\vspace{1.3mm} + +Thom computed unoriented cobordism. Oriented and complex cobordism +came later. In simplest form, the calculations use the Adams spectral +sequence: + +\vspace{1mm} + +\noindent +{\em J. Milnor. On the cobordism ring $\Omega^*$ and a complex analogue.} Amer. J. +Math. 82(1960), 505--521. + +\vspace{1mm} + +\noindent +{\em C.T.C. Wall. A characterization of simple modules over the Steenrod algebra +mod $2$.} Topology 1(1962), 249--254. + +\vspace{1mm} + +\noindent +{\em A. Liulevicius. A proof of Thom's theorem.} Comm. Math. Helv. 37(1962), 121--131. + +\vspace{1mm} + +\noindent +{\em A. Liulevicius. Notes on homotopy of Thom spectra.} Amer. J. Math. 86(1964), 1--16. + +\vspace{1.3mm} + +A very useful compendium of calculations of cobordism groups: + +\vspace{1mm} + +\noindent +{\em R. Stong. Notes on cobordism theory.} Princeton University Press. 1968. + +\section{Generalized homology theory and stable homotopy theory} + +Two classical references, the second of which also gives detailed information about complex +cobordism that is of fundamental importance to the subject. + +\vspace{1mm} + +\noindent +{\em G.W. Whitehead. Generalized homology theories.} Trans. Amer. Math. Soc. 102(1962), 227--283. + +\vspace{1mm} + +\noindent +{\em J.F. Adams. Stable homotopy and generalised homology.} Chicago Lectures in Mathematics. +University of Chicago Press. 1974. Reprinted in 1995. + +\vspace{1.3mm} + +An often overlooked but interesting book on the subject: + +\vspace{1mm} + +\noindent +{\em H.R. Margolis. Spectra and the Steenrod algebra. Modules over the Steenrod +algebra and the stable homotopy category.} North-Holland. 1983. + +\vspace{1.3mm} + +Foundations for equivariant stable homotopy theory are established in: + +\vspace{1mm} + +\noindent +{\em L.G. Lewis, Jr., J.P. May, and M.Steinberger (with contributions by +J.E. McClure). Equivariant stable homotopy theory.} Lecture Notes in +Mathematics Vol. 1213. Springer-Verlag. 1986. + +\section{Quillen model categories} + +In the introduction, I alluded to axiomatic treatments of ``homotopy theory.'' +Here are the original and two more recent references: + +\vspace{1mm} + +\noindent +{\em D.G. Quillen. Homotopical algebra.} Lecture Notes in Mathematics +Vol. 43. Springer-Verlag. 1967. + +\vspace{1mm} + +\noindent +{\em W.G. Dwyer and J. Spalinski. Homotopy theories and model categories}. +In A handbook of algebraic topology, edited by I.M. James. North-Holland. 1995. + +\vspace{1.3mm} + +The cited ``{\em Handbook}'' (over 1300 pages) contains an uneven but very interesting +collection of expository articles on a wide variety of topics in algebraic topology. + +\vspace{1.3mm} + +\noindent +{\em M. Hovey. Model categories.} Amer. Math. Soc. Surveys and Monographs No. 63. 1998. + +\section{Localization and completion; rational homotopy theory} + +Since the early 1970s, it has been standard practice in algebraic topology to localize +and complete topological spaces, and not just their algebraic invariants, at sets of primes +and then to study the subject one prime at a time, or rationally. Two of the basic original +references are: + +\vspace{1mm} + +\noindent +{\em D. Sullivan. The genetics of homotopy theory and the Adams conjecture.} +Annals of Math. 100(1974), 1--79. + +\vspace{1mm} + +\noindent +{\em A.K. Bousfield and D.M. Kan. Homotopy limits, completions, and localizations.} +Lecture Notes in Mathematics Vol. 304. Springer-Verlag. 1972. + +\vspace{1.3mm} + +A more accessible introduction to localization and a readable recent +paper on completion are: + +\vspace{1mm} + +\noindent +{\em P. Hilton, G. Mislin, and J. Roitberg. Localization of nilpotent groups +and spaces.} North-Holland. 1975. + +\vspace{1mm} + +\noindent +{\em F. Morel. Quelques remarques sur la cohomologie modulo $p$ continue des +pro-$p$-espaces et les resultats de J. Lannes concernent les espaces fonctionnel +Hom$(BV,X)$.} Ann. Sci. Ecole Norm. Sup. (4)26(1993), 309--360. + +\vspace{1.3mm} + +When spaces are rationalized, there is a completely algebraic description of the +result. The main original reference and a more accessible source are: + +\vspace{1mm} + +\noindent +{\em D. Sullivan. Infinitesimal computations in topology.} Publ. Math. +IHES 47(1978), 269--332. + +\vspace{1mm} + +\noindent +{\em A.K. Bousfield and V.K.A.M. Gugenheim. On PL de Rham theory and rational homotopy +type.} Memoirs Amer. Math. Soc. No. 179. 1976. + +\section{Infinite loop space theory} + +Another area well established by the mid-1970s. The following book is a +delightful read, with capsule introductions of many topics other than infinite +loop space theory, a very pleasant starting place for learning modern +algebraic topology: + +\vspace{1mm} + +\noindent +{\em J.F. Adams. Infinite loop spaces.} Annals of Math. Studies No. 90. Princeton +University Press. 1978. + +\vspace{1.3mm} + +The following survey article is less easy going, but gives an indication of +the applications to high dimensional geometric topology and to algebraic $K$-theory: + +\vspace{1mm} + +\noindent +{\em J.P. May. Infinite loop space theory.} Bull. Amer. Math. Soc. 83(1977), +456--494. + +\vspace{1.3mm} + +Five monographs, each containing a good deal of expository material, +that give a variety of theoretical and calculational developments and applications +in this area: + +\vspace{1mm} + +\noindent +{\em J.P. May. The geometry of iterated loop spaces.} Lecture Notes in Mathematics +Vol. 271. Springer-Verlag. 1972. + +\vspace{1mm} + +\noindent +{\em J.M. Boardman and R.M. Vogt. Homotopy invariant algebraic structures on topological +spaces.} Lecture Notes in Mathematics Vol. 347. Springer-Verlag. 1973. + +\vspace{1mm} + +\noindent +{\em F.R. Cohen, T.J. Lada, and J.P. May. The homology of iterated loop spaces.} +Lecture Notes in Mathematics Vol. 533. Springer-Verlag. 1976. + +\vspace{1mm} + +\noindent +{\em J.P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). +$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra.} Lecture Notes in +Mathematics Vol. 577. Springer-Verlag. 1977. + +\vspace{1mm} + +\noindent +{\em R. Bruner, J.P. May, J.E. McClure, and M. Steinberger. $H_{\infty}$ ring +spectra and their applications.} Lecture Notes in Mathematics Vol. 1176. +Springer-Verlag. 1986. + +\section{Complex cobordism and stable homotopy theory} + +Adams' book cited in \S16 gives a spectral sequence for the computation of stable +homotopy groups in terms of generalized cohomology theories. Starting from complex cobordism +and related theories, its use has been central to two waves of major developments in stable +homotopy theory. A good exposition for the first wave: + +\vspace{1mm} + +\noindent +{\em D.C. Ravenel. Complex cobordism and stable homotopy groups of spheres.} Academic +Press. 1986. + +\vspace{1.3mm} + +The essential original paper and a very nice survey article on the second wave: + +\vspace{1mm} + +\noindent +{\em E. Devinatz, M.J. Hopkins, and J.H. Smith. Nilpotence and stable homotopy theory.} +Annals of Math. 128(1988), 207--242. + +\vspace{1mm} + +\noindent +{\em M.J. Hopkins. Global methods in homotopy theory.} In Proceedings of the 1985 LMS +Symposium on homotopy theory, edited by J.D.S. Jones and E. Rees. +London Mathematical Society. 1987. + +\vspace{1.3mm} + +The cited {\em Proceedings} contain good introductory survey articles on several other +topics in algebraic topology. A larger scale exposition of the second wave is: + +\vspace{1mm} + +\noindent +{\em D.C. Ravenel. Nilpotence and periodicity in stable homotopy theory.} +Annals of Math. Studies No. 128. Princeton University Press. 1992. + +\section{Follow-ups to this book} + +There is a leap from the level of this introductory book to that of the most +recent work in the subject. One recent book that helps fill the gap is: + +\vspace{1mm} + +{\em P. Selick. Introduction to homotopy theory.} Fields Institute Monographs No. 9. +American Mathematical Society. 1997. + +\vspace{1.3mm} + +There is a recent expository book for the reader who would like to jump +right in and see the current state of algebraic topology; although it focuses on +equivariant theory, it contains introductions and discussions of many non-equivariant +topics: + +\vspace{1mm} + +\noindent +{\em J.P. May et al. Equivariant homotopy and cohomology theory.} +NSF-CBMS Regional Conference Monograph. 1996. + +\vspace{1.3mm} + +For the reader of the last section of this book whose appetite has been whetted for +more stable homotopy theory, there is an expository article that motivates and explains +the properties that a satisfactory category of spectra should have: + +\vspace{1mm} + +\noindent +{\em J.P. May. Stable algebraic topology and stable topological algebra.} +Bulletin London Math. Soc. 30(1998), 225--234. + +\vspace{1.3mm} + +The following monograph gives such a category, with many applications; more +readable accounts appear in the {\em Handbook} cited in \S17 and in the book just +cited: + +\vspace{1mm} + +\noindent +{\em A. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May, with an appendix by M. Cole. +Rings, modules, and algebras in stable homotopy theory.} Amer. Math. Soc. Surveys and +Monographs No. 47. 1997. + +%\input{ConciseRevised.ind} + +\end{document} diff --git a/resources/MayConcise/lacromay.sty b/resources/MayConcise/lacromay.sty new file mode 100644 index 0000000..faf2e6d --- /dev/null +++ b/resources/MayConcise/lacromay.sty @@ -0,0 +1,263 @@ +% LACROMAY.STY - Extra Math Definitions and Symbols +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{lacromay}[1998/07/22 v1.0 + Extra Math Definitions and Symbols] +%\RequirePackage{amssymb}[1995/01/01] +%% Change \lhd, \rhd to use the amssymb symbols +\renewcommand{\lhd}{\vartriangleleft} +\renewcommand{\rhd}{\vartriangleright} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\catcode`\ =9 +\endlinechar=-1 % Make things readable. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Setup to use Ralph Smith Formal Script font: +\DeclareFontFamily{OMS}{rsfs}{\skewchar\font'60} +\DeclareFontShape{OMS}{rsfs}{m}{n}{<-5>rsfs5 <5-7>rsfs7 <7->rsfs10 }{} +\DeclareSymbolFont{rsfs}{OMS}{rsfs}{m}{n} +\DeclareSymbolFontAlphabet{\scr}{rsfs} + +\let\overto\xrightarrow +\def\circle{\mathaccent"7017} + +%Kate's macros +\newcommand{\chara}{\operatorname{char}} +\newcommand{\degree}{\operatorname{deg}} +\newcommand{\Kernel}{\operatorname{Ker}} +\newcommand{\image}{\operatorname{Im}} +\newcommand{\Cokernel}{\operatorname{Coker}} + +% script letters small s then capital letter +\newcommand{\sA}{\scr{A}} +\newcommand{\sB}{\scr{B}} +\newcommand{\sC}{\scr{C}} +\newcommand{\sD}{\scr{D}} +\newcommand{\sE}{\scr{E}} +\newcommand{\sF}{\scr{F}} +\newcommand{\sG}{\scr{G}} +\newcommand{\sH}{\scr{H}} +\newcommand{\sI}{\scr{I}} +\newcommand{\sJ}{\scr{J}} +\newcommand{\sK}{\scr{K}} +\newcommand{\sL}{\scr{L}} +\newcommand{\sM}{\scr{M}} +\newcommand{\sN}{\scr{N}} +\newcommand{\sO}{\scr{O}} +\newcommand{\sP}{\scr{P}} +\newcommand{\sQ}{\scr{Q}} +\newcommand{\sR}{\scr{R}} +\newcommand{\sS}{\scr{S}} +\newcommand{\sT}{\scr{T}} +\newcommand{\sU}{\scr{U}} +\newcommand{\sV}{\scr{V}} +\newcommand{\sW}{\scr{W}} +\newcommand{\sX}{\scr{X}} +\newcommand{\sY}{\scr{Y}} +\newcommand{\sZ}{\scr{Z}} + +% Font used for operads small o then capital letter +% in case I change my mind, give the font its own name +\let\opsymbfont\mathcal + +\newcommand{\oA}{{\opsymbfont{A}}} +\newcommand{\oB}{{\opsymbfont{B}}} +\newcommand{\oC}{{\opsymbfont{C}}} +\newcommand{\oD}{{\opsymbfont{D}}} +\newcommand{\oE}{{\opsymbfont{E}}} +\newcommand{\oF}{{\opsymbfont{F}}} +\newcommand{\oG}{{\opsymbfont{G}}} +\newcommand{\oH}{{\opsymbfont{H}}} +\newcommand{\oI}{{\opsymbfont{I}}} +\newcommand{\oJ}{{\opsymbfont{J}}} +\newcommand{\oK}{{\opsymbfont{K}}} +\newcommand{\oL}{{\opsymbfont{L}}} +\newcommand{\oM}{{\opsymbfont{M}}} +\newcommand{\oN}{{\opsymbfont{N}}} +\newcommand{\oO}{{\opsymbfont{O}}} +\newcommand{\oP}{{\opsymbfont{P}}} +\newcommand{\oQ}{{\opsymbfont{Q}}} +\newcommand{\oR}{{\opsymbfont{R}}} +\newcommand{\oS}{{\opsymbfont{S}}} +\newcommand{\oT}{{\opsymbfont{T}}} +\newcommand{\oU}{{\opsymbfont{U}}} +\newcommand{\oV}{{\opsymbfont{V}}} +\newcommand{\oW}{{\opsymbfont{W}}} +\newcommand{\oX}{{\opsymbfont{X}}} +\newcommand{\oY}{{\opsymbfont{Y}}} +\newcommand{\oZ}{{\opsymbfont{Z}}} + +\DeclareMathAlphabet{\eus}{U}{eus}{m}{n} +%\SetMathAlphabet{\eus}{bold}{U}{eus}{b}{n} + +% Font used for categories small a then capital letter +% would use small c but too many conflicts with xypic +% in case I change my mind, give the font its own name +\let\catsymbfont\eus + +\newcommand{\aA}{{\catsymbfont{A}}} +\newcommand{\aB}{{\catsymbfont{B}}} +\newcommand{\aC}{{\catsymbfont{C}}} +\newcommand{\aD}{{\catsymbfont{D}}} +\newcommand{\aE}{{\catsymbfont{E}}} +\newcommand{\aF}{{\catsymbfont{F}}} +\newcommand{\aG}{{\catsymbfont{G}}} +\newcommand{\aH}{{\catsymbfont{H}}} +\newcommand{\aI}{{\catsymbfont{I}}} +\newcommand{\aJ}{{\catsymbfont{J}}} +\newcommand{\aK}{{\catsymbfont{K}}} +\newcommand{\aL}{{\catsymbfont{L}}} +\newcommand{\aM}{{\catsymbfont{M}}} +\newcommand{\aN}{{\catsymbfont{N}}} +\newcommand{\aO}{{\catsymbfont{O}}} +\newcommand{\aP}{{\catsymbfont{P}}} +\newcommand{\aQ}{{\catsymbfont{Q}}} +\newcommand{\aR}{{\catsymbfont{R}}} +\newcommand{\aS}{{\catsymbfont{S}}} +\newcommand{\aT}{{\catsymbfont{T}}} +\newcommand{\aU}{{\catsymbfont{U}}} +\newcommand{\aV}{{\catsymbfont{V}}} +\newcommand{\aW}{{\catsymbfont{W}}} +\newcommand{\aX}{{\catsymbfont{X}}} +\newcommand{\aY}{{\catsymbfont{Y}}} +\newcommand{\aZ}{{\catsymbfont{Z}}} + +% blackboard bold letters b then capital letter +%% Change \Bbb to \mathbb (for consistency with other LaTeX2e math font +%% names): +%\def\mathbb#1{\protect\text{$\protect\mathbb{#1}$}} + +% blackboard bold letters b then capital letter +%\def\mathbb#1{\protect\text{$\protect\Bbb{#1}$}} +\newcommand{\bA}{\mathbb{A}} +\newcommand{\bB}{\mathbb{B}} +\newcommand{\bC}{\mathbb{C}} +\newcommand{\bD}{\mathbb{D}} +\newcommand{\bE}{\mathbb{E}} +\newcommand{\bF}{\mathbb{F}} +\newcommand{\bG}{\mathbb{G}} +\newcommand{\bH}{\mathbb{H}} +\newcommand{\bI}{\mathbb{I}} +\newcommand{\bJ}{\mathbb{J}} +\newcommand{\bK}{\mathbb{K}} +\newcommand{\bL}{\mathbb{L}} +\newcommand{\bM}{\mathbb{M}} +\newcommand{\bN}{\mathbb{N}} +\newcommand{\bO}{\mathbb{O}} +\newcommand{\bP}{\mathbb{P}} +\newcommand{\bQ}{\mathbb{Q}} +\newcommand{\bR}{\mathbb{R}} +\newcommand{\bS}{\mathbb{S}} +\newcommand{\bT}{\mathbb{T}} +\newcommand{\bU}{\mathbb{U}} +\newcommand{\bV}{\mathbb{V}} +\newcommand{\bW}{\mathbb{W}} +\newcommand{\bX}{\mathbb{X}} +\newcommand{\bY}{\mathbb{Y}} +\newcommand{\bZ}{\mathbb{Z}} + +% Greek letters (first two letters, in small or cap; add z for variants) +\newcommand{\al}{\alpha} +\newcommand{\be}{\beta} +\newcommand{\ga}{\gamma} +\newcommand{\de}{\delta} +\newcommand{\pa}{\partial} %pretend its Greek +%\newcommand{\ep}{\epsilon} +\newcommand{\epz}{\varepsilon} +\newcommand{\ph}{\phi} +\newcommand{\phz}{\varphi} +\newcommand{\et}{\eta} +%\newcommand{\xi}{\xi} +\newcommand{\io}{\iota} +\newcommand{\ka}{\kappa} +\newcommand{\la}{\lambda} +%\newcommand{\mu}{\mu} +%\newcommand{\nu}{\nu} +\newcommand{\tha}{\theta} +\newcommand{\thz}{\vartheta} +%\newcommand{\pi}{\pi} +\newcommand{\rh}{\rho} +\newcommand{\si}{\sigma} +\newcommand{\ta}{\tau} +\newcommand{\ch}{\chi} +\newcommand{\ps}{\psi} +\newcommand{\ze}{\zeta} +\newcommand{\om}{\omega} +\newcommand{\GA}{\Gamma} +\newcommand{\LA}{\Lambda} +\newcommand{\DE}{\Delta} +\newcommand{\SI}{\Sigma} +\newcommand{\THA}{\Theta} +\newcommand{\OM}{\Omega} +\newcommand{\XI}{\Xi} +\newcommand{\UP}{\Upsilon} +\newcommand{\PI}{\Pi} +\newcommand{\PS}{\Psi} +\newcommand{\PH}{\Phi} + +% preserve old meaning of \ep when outside of math mode +%\let\old@ep=\ep +%\def\ep{\ifmmode\epsilon\else\old@ep\fi} +% symbols --- three letter commands +\newcommand{\com}{\circ} % composition of functions +\newcommand{\iso}{\cong} % preferred isomorphism symbol +\newcommand{\htp}{\simeq} % homotopy symbol +\newcommand{\ten}{\otimes} % tensor product +\newcommand{\add}{\oplus} % direct sum +\newcommand{\thp}{\ltimes} % twisted half-smash product +\newcommand{\sma}{\wedge} % smash product +\newcommand{\wed}{\vee} % wedge sum + +\newcommand{\ef}{\text{$E_\infty\ $}} +\newcommand{\af}{\text{$A_\infty\ $}} + +\newcommand{\ul}{\underline} + +%\gdef\overto#1{{\buildrel{#1}\over\longrightarrow}} +\newcommand{\overfrom}[1]{\xleftarrow{#1}} + +\newcommand{\tand}{\text{\ \ and \ \ }} %``and'' between formulas in display + +\newcommand{\ip}[1]{\text{$\left\langle#1\right\rangle$}} % inner product +\newcommand{\rtarr}{\longrightarrow} +\newcommand{\ltarr}{\longleftarrow} +\newcommand{\from}{\longleftarrow} +\newcommand{\monoto}{\lhook\joinrel\relbar\joinrel\rightarrow} +\newcommand{\epito}{\relbar\joinrel\twoheadrightarrow} + +% operators +\def\quickop#1{\expandafter\newcommand\csname #1\endcsname{\operatorname{#1}}} +\quickop{Hom} \quickop{End} \quickop{Aut} \quickop{Tel} \quickop{Mic} +\quickop{Ext} \quickop{Tor} \quickop{Id} \quickop{Coker} \quickop{Ker} +\quickop{Lim} \quickop{Colim} \quickop{Holim} \quickop{Hocolim} +\quickop{id} \quickop{tel} \quickop{mic} \quickop{coker} +\quickop{colim} \quickop{holim} \quickop{hocolim} \quickop{im} + +% \limit --- lim sub right arrow +\let\limit=\varinjlim +% \colimit --- lim sub left arrow +\let\colimit=\varprojlim + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SETS - the macro \set and \sset +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% \sset, or ``singleton set'' denotes a set of the form \{ x \}. +% An optional argument is the subscript. This might typically used +% for giving an indexing set. +\newtoks\sset@tok +\newcommand{\sset}[1]{\sset@tok={#1}\futurelet\sset@temp\sset@action} +\def\sset@witharg[#1]{\text{$\left\{\the\sset@tok\right\}_{#1}$}} +\def\sset@withoutarg{\text{$\left\{\the\sset@tok\right\}$}} +\def\sset@action{\ifx\sset@temp[%] +\let\sset@next=\sset@witharg\else\let\sset@next=\sset@withoutarg\fi\sset@next} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% End readability. +\catcode`\ =10 \endlinechar=`\^^M +\endinput % of macromay.sty + + + + + + diff --git a/src/HottBook/Chapter2.lagda.md b/src/HottBook/Chapter2.lagda.md index 359dd72..5e76081 100644 --- a/src/HottBook/Chapter2.lagda.md +++ b/src/HottBook/Chapter2.lagda.md @@ -738,31 +738,55 @@ open axiom2∙10∙3 ### Theorem 2.11.1 ``` --- theorem2∙11∙1 : {A B : Set} --- → (eqv @ (f , f-eqv) : A ≃ B) --- → (a a' : A) --- → (a ≡ a') ≃ (f a ≡ f a') --- theorem2∙11∙1 (f , f-eqv) a a' = --- let --- open ≡-Reasoning --- mkQinv g α β = isequiv-to-qinv f-eqv --- inv : (f a ≡ f a') → a ≡ a' --- inv p = (sym (β a)) ∙ (ap g p) ∙ (β a') --- backward : (p : f a ≡ f a') → (ap f ∘ inv) p ≡ id p --- backward q = begin --- ap f ((sym (β a)) ∙ (ap g q) ∙ (β a')) ≡⟨ lemma2∙2∙2.i (sym (β a)) (ap g q ∙ β a') ⟩ --- ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩ --- ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩ --- id q ∎ --- forward : (p : a ≡ a') → (inv ∘ ap f) p ≡ id p --- forward p = begin --- (sym (β a)) ∙ (ap g (ap f p)) ∙ (β a') ≡⟨ ap (λ p → (sym (β a)) ∙ p ∙ (β a')) (lemma2∙2∙2.iii f g p) ⟩ --- (sym (β a)) ∙ (ap (g ∘ f) p) ∙ (β a') ≡⟨ {! !} ⟩ --- (sym (β a)) ∙ (ap id p) ∙ (β a') ≡⟨ {! !} ⟩ --- id p ∎ --- eqv = mkQinv inv backward forward --- in --- ap f , qinv-to-isequiv eqv +theorem2∙11∙1 : {A B : Set} + → (eqv @ (f , f-eqv) : A ≃ B) + → (a a' : A) + → (a ≡ a') ≃ (f a ≡ f a') +theorem2∙11∙1 (f , f-eqv) a a' = + let + open ≡-Reasoning + mkQinv g α β = isequiv-to-qinv f-eqv + inv : (f a ≡ f a') → a ≡ a' + inv p = (sym (β a)) ∙ (ap g p) ∙ (β a') + backward : (p : f a ≡ f a') → (ap f ∘ inv) p ≡ id p + backward q = begin + ap f ((sym (β a)) ∙ (ap g q) ∙ (β a')) ≡⟨ lemma2∙2∙2.i (sym (β a)) (ap g q ∙ β a') ⟩ + ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩ + ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩ + id q ∎ + forward : (p : a ≡ a') → (inv ∘ ap f) p ≡ id p + forward p = begin + (sym (β a)) ∙ (ap g (ap f p)) ∙ (β a') ≡⟨ ap (λ p → (sym (β a)) ∙ p ∙ (β a')) (lemma2∙2∙2.iii f g p) ⟩ + (sym (β a)) ∙ (ap (g ∘ f) p) ∙ (β a') ≡⟨ {! !} ⟩ + -- (sym (β a)) ∙ (ap id p) ∙ (β a') ≡⟨ {! !} ⟩ + id p ∎ + eqv = mkQinv inv backward forward + in + ap f , qinv-to-isequiv eqv +``` + +### Theorem 2.11.2 + +``` +-- module theorem2∙11∙2 where +-- i : {A : Set} {a x1 x2 : A} +-- → (p : x1 ≡ x2) +-- → (q : a ≡ x1) +-- → transport (λ y → a ≡ y) p q ≡ q ∙ p +-- i {A} {a} {x1} {x2} p q = +-- J (λ x3 x4 p1 → (q1 : a ≡ x3) → transport (λ y → a ≡ y) p1 q1 ≡ q1 ∙ p1) +-- (λ x3 q1 → J (λ x5 x6 q2 → transport (λ y → a ≡ y) refl q1 ≡ q1 ∙ refl) (λ x4 → {! refl !}) a x3 q1) +-- x1 x2 p q +``` + +### Theorem 2.11.3 + +``` +theorem2∙11∙3 : {A B : Set} → {f g : A → B} → {a a' : A} + → (p : a ≡ a') + → (q : f a ≡ g a) + → transport (λ x → f x ≡ g x) p q ≡ sym (ap f p) ∙ q ∙ (ap g p) +theorem2∙11∙3 p q = {! !} ``` ## 2.12 Coproducts diff --git a/src/HottBook/Chapter6.lagda.md b/src/HottBook/Chapter6.lagda.md index 3a0e4e3..280eb94 100644 --- a/src/HottBook/Chapter6.lagda.md +++ b/src/HottBook/Chapter6.lagda.md @@ -3,6 +3,7 @@ module HottBook.Chapter6 where open import HottBook.Chapter1 open import HottBook.Chapter2 +open import HottBook.Chapter2Lemma221 ``` # 6 Higher inductive types @@ -10,12 +11,14 @@ open import HottBook.Chapter2 ### Definition 6.2.2 (Dependent paths) ``` -dep-path : {A : Set} +definition6∙2∙2 : {A : Set} → (P : A → Set) → {x y : A} → (p : x ≡ y) → (u : P x) → (v : P y) → Set -dep-path P p u v = transport P p u ≡ v +definition6∙2∙2 P p u v = transport P p u ≡ v + +syntax definition6∙2∙2 P p u v = u ≡[ P , p ] v ``` Circle definition @@ -27,28 +30,34 @@ postulate loop : base ≡ base S¹-elim : (P : S¹ → Set) → (p-base : P base) - → (p-loop : dep-path P loop p-base p-base) + → (p-loop : p-base ≡[ P , loop ] p-base) → (x : S¹) → P x ``` ### Lemma 6.2.5 ``` -lemma6∙2∙5 : {A : Set} - → (a : A) - → (p : a ≡ a) - → S¹ → A +lemma6∙2∙5 : {A : Set} → (a : A) → (p : a ≡ a) → S¹ → A lemma6∙2∙5 {A} a p circ = S¹-elim P p-base p-loop circ where P : S¹ → Set P _ = A + p-base : P base p-base = a - p-loop : transport P loop a ≡ a - p-loop = - let wtf = lemma2∙3∙8 (λ x → {! !}) loop in - {! !} + p-loop : a ≡[ P , loop ] a + p-loop = transportconst A loop a ∙ p +``` + +### Lemma 6.2.8 + +``` +lemma6∙2∙8 : {A : Set} {f g : S¹ → A} + → (p : f base ≡ g base) + → (q : (ap f loop) ≡[ (λ x → x ≡ x) , p ] (ap g loop)) + → (x : S¹) → f x ≡ g x +lemma6∙2∙8 {A} {f} {g} p q = S¹-elim (λ x → f x ≡ g x) p {! !} ``` ## 6.3 The interval