From c9758ba4a21d0d77a63e73d06b29d78bd72ec231 Mon Sep 17 00:00:00 2001 From: Egbert Rijke Date: Fri, 4 Dec 2015 16:05:00 -0500 Subject: [PATCH] removing K-theory part --- Notes/notes.tex | 243 ------------------------------------------------ 1 file changed, 243 deletions(-) diff --git a/Notes/notes.tex b/Notes/notes.tex index 4b1ed1e..2468c71 100644 --- a/Notes/notes.tex +++ b/Notes/notes.tex @@ -394,247 +394,4 @@ a filtration $0\subseteq F^0_n\subseteq\cdots\subseteq F^n_n=H_n(X;G)$ of ... \end{enumerate} \end{thm} -\part{K-theory} - -\section{Vector bundles} - -\subsection{Basic spaces} -\begin{defn} -The \define{$n$-sphere} $\Sn^n$ is the subspace of $\R^{n+1}$ consisting of unit vectors. -The \define{real projective $n$-space} $\R P^n$ is the space of lines in -$\R^{n+1}$ through the origin. Equivalently, we may regard $\R P^n$ as the quotient -space of $\Sn^n$ in which the antipodal pairs of points are identified. Notice -that $\R P^1\approx \Sn^1$. -\end{defn} - -\begin{defn} -For each $n$, we may include the $n$-sphere $\Sn^n$ into $\Sn^{n+1}$ by mapping -it into the equator. These inclusions induce inclusions $\R P^n\to \R P^{n+1}$. -We define $\R P^\infty$ to be the sequential colimit of $\R P^n$. -\end{defn} - -\subsection{Definition and basic properties} -\begin{defn} -An \define{$n$-dimensional vector bundle} is a map $p:E\to B$ together with a -real vector space structure on $p^{-1}(b)$ for each $b\in B$, satisfying the -\define{local triviality condition}, which says that there is an open cover -$\mathcal{C}$ of $B$, with homeomorphisms $h_U:p^{-1}(U)\to U\times\mathbb{R}^n$ -for each $U\in\mathcal{C}$, which maps $p^{-1}(b)$ to $\{b\}\times\mathbb{R}^n$ -for each $b\in U$. - -The functions $h_U$ are also called \define{local trivializations}. Given a -vector bundle $p:E\to B$, the space $B$ is called the \define{base space}, the -space $E$ is called the \define{total space}, and the spaces $p^{-1}(b)$ are -called the \define{fibers}. A $1$-dimensional -vector bundle is also called a \define{line bundle}. -\end{defn} - -\begin{defn} -An \define{isomorphism of vector bundles} from $p:E\to B$ to $p':E'\to B$ -consists of a map $h:E\to E'$ satisfying $p'\circ h=p$, -which induces a linear isomorphism -$p^{-1}(b)\to p'^{-1}(b)$ between each of the fibers. -\end{defn} - -\begin{lem} -If $h:E\to E'$ is an isomorphism of vector bundles, then the underlying map -of type $E\to E'$ is a homeomorphism. -\end{lem} - -\begin{proof} -Suppose $h:E\to E'$ induces isomorphisms $p^{-1}(b)\to p'^{-1}(b)$ for each -$b\in B$. Then, for each $x\in E'$ we have an isomorphism from -$p^{-1}(p'(x))$ to $p'^{-1}(p'(x))$. Since $x\in p'^{-1}(p'(x))$, we find -an element $y\in p^{-1}(p'(x))\subseteq E$. Thus, $h$ is surjective. Now suppose -that $x,x'\in E$ are two elements for which $h(x)=h(x')$. Since $p'\circ h=p$, -it follows that $x'\in p^{-1}(x)$. Now, the fact that $h$ induces an isomorphism -between fibers implies that $x=x'$. - -Thus, $h$ has an inverse function $k:E'\to E$, and we need to show that this -function is continuous. It suffices to show that $k|_U$ is continuous for each -$U$ on which $p'$ is trivial. Let $x\in B$, and compose the map -$h_U:p^{-1}(U)\to p'^{-1}(U)$ with its local trivializations. Thus, we obtain -a map $g_U:U\times\R^n\to U\times \R^n$, mapping $(x,y)$ to $(x,A(y))$, where -$A$ is a linear isomorphism. -\end{proof} - -In the following definition, we give a vector bundle by a gluing construction. - -\begin{defn} -Consider a space $B$, and an open cover $\mathcal{C}$ which is closed under -finite intersections. Then $\mathcal{C}$ may be considered a poset ordered by -inclusion. - -A \define{collection of gluing functions} consists of a continuous choice of linear -isomorphisms $g_{U,V}:U\cap V\to GL_n(\R)$ satisfying the \define{cocycle -condition} -\begin{equation*} -g_{V,W}\circ g_{U,V}=g_{U,W} -\end{equation*} -on $U\cap V\cap W$, for every -$U,V,W\in\mathcal{C}$. Such a collection of gluing functions determines a functor -$\mathcal{C}\to\mathbf{Top}$, which is given on points by $U\mapsto U\times\R^n$, -and on morphisms by $(x,v)\mapsto(x,A(v))$, for each $U\subseteq V$ determining -a linear isomorphism $A$. - -The colimit of this functor is the total space of a vector bundle. -\end{defn} - -\begin{eg} -There are lots of examples of vector bundles: -\begin{enumerate} -\item The \define{$n$-dimensional trivial bundle} over $B$ is defined to be -$\proj1:B\times\mathbb{R}^n\to B$. So the trivial bundle is the one which is -\emph{globally} trivial. We will write the $n$-dimensional trivial bundle over -$B$ as $\epsilon^n\to B$. -\item The circle may be regarded as the quotient of $[0,1]$ modulo the end points. -The \define{Mobius bundle} is the line bundle over $\Sn^1$ to have total space -$E\defeq [0,1]\times\R$, with the identifications $(0,t)\sim(1,-t)$. -\item The \define{tangent bundle} of the unit sphere $\Sn^n$, viewed as a subspace of -$\R^{n+1}$, is defined to be the subspace $E\defeq\{(x,v)\in\Sn^n\times\R^{n+1} -\mid x\perp v\}$ of $\R^{2n+2}$, which projects onto $\Sn^n$. - -The $n$-sphere is covered by $2n+2$ open hemispheres, centering at $\pm e_i$, -where $e_i\in\R^{n+1}$ is a basis vector. -\item The \define{normal bundle} of the unit sphere $\Sn^n$ is the line bundle -with $E$ consisting of pairs $(x,v)\in\Sn^n\times\R^{n+1}$ such that $v=tx$ for -some $t\in\R$. \emph{The normal bundle on $\Sn^n$ is isomorphic to the trivial line -bundle $\Sn^n\times\R\to\Sn^n$.} -\item The \define{canonical line bundle} $p:E\to \R P^n$ has as its total space -the subspace $E\subseteq \R P^{n+1}\times\R ^{n+1}$ consisting of pairs -$(l,v)$ with $v\in l$. \emph{The M\"obius line bundle is isomorphic to the -canonical line bundle on $\Sn^1$.} -\item The inclusions $\R P^n\subseteq \R P^{n+1}$ induce inclusions of the -canonical line bundles. The sequential colimit of the canonical line bundles -produces the canonical line bundle on $\R P^\infty$. -\end{enumerate} -\end{eg} - -\begin{defn} -Given two vector bundles $p:E\to B$ and $p':E'\to B$ over the same base space -$B$, we obtain a vector bundle $p\oplus p': E\oplus E'\to B$, fitting in the -pullback square -\begin{equation*} -\begin{tikzcd} -E\oplus E' \arrow[r] \arrow[d] \arrow[dr,"{p\oplus p'}" description ] & E' \arrow[d,"{p'}"] \\ -E \arrow[r,swap,"p"] & B -\end{tikzcd} -\end{equation*} -\end{defn} - -\begin{eg} -\begin{enumerate} -\item The direct sum of the tangent and normal bundles on $\Sn^n$ is the trivial -bundle $\Sn^n\times\R^{n+1}$. -\end{enumerate} -\end{eg} - -\begin{defn} -Let $p:E\to B$ and $p':E'\to B$ be two vector bundles over the same space $B$, -and choose an open cover $\mathcal{C}$ such that both $E$ and $E'$ are locally -trivial with respect to $\mathcal{C}$. We define $E\otimes E'$ by gluing. - -Then we can define, for each $U,V\in\mathcal{C}$ satisfying $U\subseteq V$, we -have linear isomorphisms $g_{U,V}(x):\R^n\to\R^n$ and $g'_{U,V}(x):\R^m\to\R^m$, -induced by the local trivializations of $E$ and $E'$ respectively. These give -gluing functions $g_{U,V}(x)\otimes g'_{U,V}(x):\R^n\otimes\R^m\to\R^n\otimes\R^m$ -for each $x\in U$, and these gluing functions satisfy the cocycle condition. - -Thus, we obtain a vector bundle $E\otimes E'$ from these gluing functions. -\end{defn} - -\begin{lem} -The tensor product of vector bundles over a fixed base space is commutative, -associative, it has an identity element (the trivial bundle), and it is -distributive with respect to direct sum. -\end{lem} - -Change of base $f:B'\to B$ turns a vector bundle $E$ over $B$ to a vector -bundle $f^\ast(E)$ over $B'$. - -\begin{lem} -For any two vector bundles $E$ and $E'$ over $B$, and any $f:B'\to B$, we have -natural isomorphisms $f^\ast(E\oplus E')\approx f^\ast(E)\oplus f^\ast(E')$, and -$f^\ast(E\otimes E')\approx f^\ast(E)\otimes f^\ast(E')$. Moreover, if $f$ -is homotopic to $g$, then $f^\ast=g^\ast$. -\end{lem} - -\subsection{K-theory} - -\begin{defn} -Two vector bundles $E\to B$ and $E'\to B$ are callec \define{stably isomorphic}, -if there is an $n$ for which $E\oplus\epsilon^n\approx E'\oplus\epsilon^n$, and -we write $E\approx_s E'$ if $E$ and $E'$ are stably isomorphic. Also, -we will define the relation $E\sim E'$ if there are $m$ and $n$ such that -$E\oplus\epsilon^m\approx E'\oplus^n$. -\end{defn} - -\begin{lem} -The direct sum preserves both $\approx_s$ and $\sim$. Moreover, if $B$ is compact, -then the set of ${\sim}$-equivalence classes of vector bundles forms an abelian -group, called $\tilde{K}(B)$. If $B$ is pointed, then the tensor product turns -$\tilde{K}(B)$ into a ring. -\end{lem} - -\begin{lem} -The direct sum satisfies the cancellation property with respect to $\approx_s$, -i.e.~we have that $E\oplus E'\approx_s E\oplus E''$ implies $E'\oplus E''$. -Thus, if we define two pairs $(E,F)$ and $(E',F')$ to be equivalent to each -other whenever $E\oplus F'=E'\oplus F$, we obtain an abelian group $K(B)$ for -any compact space $B$. The tensor product turns $K(B)$ into a ring. -\end{lem} - -\begin{lem} -We have a ring isomorphism -\begin{equation*} -K(B)\approx \tilde{K}(B)\oplus\Z. -\end{equation*} -\end{lem} - -Both $K$ and $\tilde{K}$ are contravariant functors. - -\begin{lem} -If $X$ is compact Hausdorff and $A\subseteq X$ is a closed subspace, then the -inclusion and quotient maps $A\stackrel{i}{\to}X\stackrel{q}{\to}X/A$ induces -an sequence -\begin{equation*} -\begin{tikzcd} -\tilde{K}(X/A) \arrow[r,"q^\ast"] & \tilde{K}(X) \arrow[r,"i^\ast"] & \tilde{K}(A) -\end{tikzcd} -\end{equation*} -which is exact at $\tilde{K}(X)$. -\end{lem} - -\begin{lem} -If $A$ is contractible, the quotient map $q:X\to X/A$ induces a bijection -$q^\ast:\mathrm{Vect}^n(X/A)\to\mathrm{Vect}^n(X)$. -\end{lem} - -Apparently, this gives a long exact sequence of $\tilde{K}$-groups: -\begin{equation*} -\begin{tikzcd}[column sep=small] -\cdots\arrow[r] & \tilde{K}(\Sn(X)) \arrow[r] & \tilde{K}(\Sn(A)) \arrow[r] -& \tilde{K}(X/A) \arrow[r] & \tilde{K}(X) \arrow[r] & \tilde{K}(A). -\end{tikzcd} -\end{equation*} -Still considering pointed spaces, we may consider the long exact sequence of the pair $(X\times Y, -X\vee Y)$. Recall that $(X\times Y)/(X\vee Y)$ is the smash product -$X\wedge Y$, i.e.~the smash product is the pushout of $\unit\leftarrow -X\vee Y\rightarrow X\times Y$. The long exact sequence of the pair -$(X\times Y,X\vee Y)$ looks as follows: -\begin{equation*} -\begin{tikzcd}[column sep=.8em] -\cdots\arrow[r] & \tilde{K}(\Sn(X\times Y)) \arrow[r] & \tilde{K}(\Sn(X\vee Y)) \arrow[r] -& \tilde{K}(X\wedge Y) \arrow[r] & \tilde{K}(X\times Y) \arrow[r] & \tilde{K}(X\vee Y). -\end{tikzcd} -\end{equation*} - -\subsection{Bott periodicity} - -\begin{defn} -We define an \define{external product} $\mu:K(X)\otimes K(Y)\to K(X\times Y)$, -by $\mu(a\otimes b)\defeq \proj1^\ast(a)\cdot\proj2^\ast(b)$. -\end{defn} - - \end{document}