From 99c730b0a54c4996c7a2b0cb8c006f81f3d374bc Mon Sep 17 00:00:00 2001 From: Egbert Rijke Date: Tue, 8 Dec 2015 16:17:38 -0500 Subject: [PATCH] added Floris his notes --- Notes/sss.tex | 412 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 412 insertions(+) create mode 100644 Notes/sss.tex diff --git a/Notes/sss.tex b/Notes/sss.tex new file mode 100644 index 0000000..a221f4f --- /dev/null +++ b/Notes/sss.tex @@ -0,0 +1,412 @@ +\documentclass{article} + +\usepackage[utf8x]{inputenc} +\usepackage[english]{babel} +\usepackage{hyperref} + +\usepackage{amsmath,amsfonts,amsthm,amssymb,enumerate,fullpage,tikz} +\setlength{\parindent}{0pt} +\usepackage{listings} +\usepackage{color} +\newcommand{\mb}[1]{\mathbb{#1}} +\newcommand{\mc}[1]{\mathcal{#1}} +\newcommand{\Z}{\mb{Z}} +\newcommand{\N}{\mb{N}} +\newcommand{\R}{\mb{R}} +\newcommand{\C}{\mb{C}} +\newcommand{\cproj}{\C\textup{P}^\infty} +\DeclareMathOperator{\myker}{ker} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\Tor}{Tor} +\usetikzlibrary{arrows} + + +%% Theorem environment declarations (using amsthm): +\newtheorem{theorem}{Theorem}[section] +\newtheorem{axiom}[theorem]{Axiom} +\newtheorem{fact}[theorem]{Fact} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{convention}[theorem]{Convention} +\newtheorem{examplex}[theorem]{Example} +\newenvironment{example} + {\pushQED{\qed}\renewcommand{\qedsymbol}{$\triangle$}\examplex} + {\popQED\endexamplex} +\newtheorem{examples}[theorem]{Examples} +\newtheorem{notation}[theorem]{Notation} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{idea}[theorem]{Idea} + + +\begin{document} + +\title{Applications of the Serre Spectral Sequence} +\author{Floris van Doorn} +\date{November 10, 2015} +\maketitle + +\section{Serre Spectral Sequence} + +\begin{definition} +A \emph{Spectral Sequence} is a sequence $(E_{p,q}^r,d_r)$ consisting of +\begin{itemize} +\item An $R$-module $E^r_{p,q}$ for $p,q\in \Z$ and $r\geq 0$. +\item Differentials $d_{p,q}^r : E_{p,q}^r\to E_{p-r,q+r-1}^r$ such that $d_r^2=0$ +\end{itemize} +where $E^{r+1}$ is defined to be the homology of $(E^r,d^r)$. That is, +$E_{p,q}^{r+1}=\myker(d_{p,q}^r)/\im(d_{p+r,q-r+1}^r)$. The variable $r$ is called the \emph{page}, $p$ the \emph{filtration degree}, $q$ the \emph{complementary degree} and $p+q$ the \emph{total degree}. +\end{definition} + +\begin{theorem}[Serre Spectral Sequence] +Let $F \to X \twoheadrightarrow B$ be a fibration such that $B$ is path-connected and $\pi_1(B)$ +acts trivially on $H_*(F;G)$. Then +$$H_p(B;H_q(F;G))\ \Longrightarrow\ H_{p+q}(X;G).$$ This means that there is a spectral sequence +$(E_{p,q}^r,d_r)$ where $E_{p,q}^2\simeq H_p(B;H_q(F;G))$ and there is a filtration $0 \subseteq +F_{p+q}^0 \subseteq \cdots \subseteq F_{p+q}^{p+q}=H_{p+q}(X;G)$ such that $E_{p,q}^\infty\simeq +F_{p+q}^p/F_{p+q}^{p-1}$. +\end{theorem} + +Note that if $B$ is simply connected, then conditions of the theorem are satisfied. + + +\subsection{Examples} + +\begin{example} +Suppose $X=B\times F$, where $B$ is path-connected, and suppose that $G$ is a field. Then $\pi_1(B)$ +acts trivially on $H_*(F;G)$ and we have +\begin{align*} +H_n(X;G) &= \bigoplus_{p+q=n}H_p(B;G)\otimes H_q(F;G)&&\text{(K\"unneth formula)}\\ +&=\bigoplus_{p+q=n}H_p(B;H_q(F;G))&&\text{(Univ. Coeff. Th. for homology)}\\ +\end{align*} +This means that all entries in the second page survive until page infinity. The other extreme is if +$X$ is contractible, where almost nothing will survive, as we will see in the next examples. +\end{example} + +In the next example, we will use that $S^1=K(\Z,1)$ and $\cproj=K(\Z,2)$. + +\begin{example} +Consider the path space fibration of $B=\cproj$, that is $\Omega B\to PB \twoheadrightarrow B$ and +note that $S^1=\Omega B$. Since $B$ is simply connected, we can apply the Serre Spectral Sequence +with coefficients in $\Z$. We know that $E_{p,q}^2\simeq H_p(B;H_q(S^1))$ and $H_q(S^1)=0$ for $q>1$ +and $\Z$ for $q=0,1$. This means that the page $E^2$ looks like this. + +\begin{center} +\begin{tikzpicture}[>=stealth',auto,node distance=1.5cm, + main node/.style={font=\sffamily\bfseries},text height=1.5ex] + \node[main node] (-11)at (-1.2,0.8) {\mbox{}}; + \node[main node] (00) at (0,0) {$\Z$}; + \node[main node] (01) at (0,1) {$\Z$}; + \node[main node] (10) at (1.5,0) {$H_1(B)$}; + \node[main node] (11) at (1.5,1) {$H_1(B)$}; + \node[main node] (20) at (3,0) {$H_2(B)$}; + \node[main node] (21) at (3,1) {$H_2(B)$}; + \node[main node] (30) at (4.5,0) {$H_3(B)$}; + \node[main node] (31) at (4.5,1) {$H_3(B)$}; + \node[main node] (40) at (6,0) {$H_4(B)$}; + \node[main node] (41) at (6,1) {$H_4(B)$}; + \node[main node] (50) at (7.5,0) {$\cdots$}; + \node[main node] (51) at (7.5,1) {$\cdots$}; + \node at (-1.2,1) {1}; + \node at (-1.2,0) {0}; + \node at (0,-1) {0}; + \node at (1.5,-1) {1}; + \node at (3,-1) {2}; + \node at (4.5,-1) {3}; + \node at (6,-1) {4}; + \node at (7.3,-0.7) {$p$}; + \node at (-0.8,1.5) {$q$}; + \draw (-0.5,-0.5) -- (8,-0.5); + \draw (-0.5,-0.5) -- (-0.5,1.5); + \path[every node/.style={font=\sffamily\small}] +% (10) edge [->] (-11) + (20) edge [->] (01) + (30) edge [->] (11) + (40) edge [->] (21) + (50) edge [->] (31); +\end{tikzpicture} +\end{center} + +Moreover, we have $E_{p,q}^\infty=\Z$ for $p=q=0$ and $0$ otherwise. From this we can conclude that +$$H_i(\cproj,\Z)=\begin{cases} \Z & \text{if $i$ even} \\ 0 & \text{if $i$ odd.}\end{cases}$$ +\end{example} + +\begin{example} +In this example we will compute the homology groups of the loop space of the sphere, $\Omega S^n$ +for $n\geq 2$. We use the fibration $\Omega S^n \to PS^n \twoheadrightarrow S^n$ and we can apply the Serre +Spectral Sequence, since $S^n$ is simply connected. Now $H_p(S^n;G)=G$ for $p=0,n$ and $0$ +otherwise. This means that only the $0$ and the $n$ column can be nonzero. + +\begin{center} +\begin{tikzpicture}[>=stealth',auto,node distance=1.5cm, + main node/.style={font=\sffamily\bfseries},text height=1.5ex] + \node[main node] (00) at (0,0) {$\Z$}; + \node[main node] (10) at (2.5,0) {$\Z$}; + \node[main node] (01) at (0,1) {$H_{n-1}(\Omega S^n)$}; + \node[main node] (11) at (2.5,1) {$H_{n-1}(\Omega S^n)$}; + \node[main node] (02) at (0,2) {$H_{2n-2}(\Omega S^n)$}; + \node[main node] (12) at (2.5,2) {$H_{2n-2}(\Omega S^n)$}; + \node[main node] (03) at (0,3) {$H_{3n-3}(\Omega S^n)$}; + \node[main node] (13) at (2.5,3) {$H_{3n-3}(\Omega S^n)$}; + \node[main node] (04) at (0,4) {$\vdots$}; + \node[main node] (14) at (2.5,4) {$\vdots$}; + \node at (-1.9,0) {0}; + \node at (-1.9,1) {$n-1$}; + \node at (-1.9,2) {$2n-2$}; + \node at (-1.9,3) {$3n-3$}; + \node at (0,-1) {0}; + \node at (2.5,-1) {$n$}; + \node at (2.8,-0.7) {$p$}; + \node at (-1.5,4.5) {$q$}; + \draw (-1.2,-0.5) -- (3,-0.5); + \draw (-1.2,-0.5) -- (-1.2,5); + \path[every node/.style={font=\sffamily\small}] + (10) edge [->] (01) + (11) edge [->] (02) + (12) edge [->] (03) + (13) edge [->] (04); +\end{tikzpicture} +\end{center} +After some reasoning, we get that $H_i(\Omega S^n,\Z)=\begin{cases} \Z & \text{for $n-1\mid i$}\\ 0 & \text{otherwise.}\end{cases}$ +\end{example} + + +\section{Serre Class Theorem} + +\begin{definition} +We say that a space $X$ is \emph{abelian} if the action of $\pi_1(X)$ on $\pi_n(X)$ is trivial for all $n\geq 1$. +\end{definition} +Note that every simply connected space is abelian. +\begin{definition} + A \emph{Serre Class} is a class $\mc C$ of abelian groups containing the trivial group such that for every SES $0\to A\to B\to C\to 0$ we have $B\in\mc C$ iff $A,C\in \mc C$. In this document I call a Serre class \emph{nice} if for every $A,B\in\mc C$ also $A\otimes B$ and $\Tor(A,B)$ are in $\mc C$. (this name is made up by me) +\end{definition} +\begin{lemma} + The following classes are nice Serre classes. +\begin{itemize} +\item $\mc {FG}$, the class of finitely generated abelian groups +\item $\mc{T}_P$ for some set $P$ of primes. This is the class of torsion abelian groups whose + elements have orders divisible only by primes in $P$. +\item $\mc{F}_P$, the finite groups in $\mc{T}_P$. +\end{itemize} +\end{lemma} +Note that $P$ is the set of all primes $\mc{T}_P$ becomes the class of all torsion abelian groups +and $\mc{F}_P$ becomes the class of all finite abelian groups. + +\begin{theorem} + Let $X$ be a path-connected and abelian space, and let $\mc C$ be a nice Serre class. Then + $$\forall(n>0)(\pi_n(X)\in\mc{C})\quad \longleftrightarrow\quad \forall(n>0)(H_n(X)\in\mc{C})$$ +\end{theorem} + +\begin{corollary} +The homotopy groups of a finite simply connected CW-complex are finitely generated. In particular, the homotopy groups of spheres are finitely generated. +\end{corollary} + +Recall the following definition and theorem. +\begin{definition} +The \emph{Hurewicz homomorphism} is the homomorphism $h : \pi_n(X) \to H_n(X)$ defined by $h([f])=f_*(\gamma)$, where $\gamma$ is a generator of $H_n(S^n)\simeq\Z$. +\end{definition} + +\begin{theorem}[Hurewicz] +Let $n\geq2$ and $X$ a $(n-1)$-connected space. Then $\widetilde H_i(X)=0$ for $i=stealth',auto,node distance=1.5cm, + main node/.style={font=\sffamily\bfseries},text height=1.5ex] + \node[main node] (-11)at (-1.2,0.8) {\mbox{}}; + \node[main node] (00) at (0,0) {$\Z x_0$}; + \node[main node] (01) at (0,1) {$\Z a$}; + \node[main node] (10) at (1.5,0) {$0$}; + \node[main node] (11) at (1.5,1) {$0$}; + \node[main node] (20) at (3,0) {$\Z x_2$}; + \node[main node] (21) at (3,1) {$\Z ax_2$}; + \node[main node] (30) at (4.5,0) {$0$}; + \node[main node] (31) at (4.5,1) {$0$}; + \node[main node] (40) at (6,0) {$\Z x_4$}; + \node[main node] (41) at (6,1) {$\Z ax_4$}; + \node[main node] (50) at (7.5,0) {$\cdots$}; + \node[main node] (51) at (7.5,1) {$\cdots$}; + \node at (-1.2,1) {1}; + \node at (-1.2,0) {0}; + \node at (0,-1) {0}; + \node at (1.5,-1) {1}; + \node at (3,-1) {2}; + \node at (4.5,-1) {3}; + \node at (6,-1) {4}; + \node at (7.3,-0.7) {$p$}; + \node at (-0.8,1.5) {$q$}; + \draw (-0.5,-0.5) -- (8,-0.5); + \draw (-0.5,-0.5) -- (-0.5,1.5); + \path[every node/.style={font=\sffamily\small}] +% (10) edge [->] (-11) + (01) edge [->] (20) + (21) edge [->] (40); +\end{tikzpicture} +\end{center} +All arrows are isomorphisms. We may assume that $d_2a=x_2$. Then we compute $d_2(ax_{2i})=x_2x_{2i}$ so we may assume that $x_2x_{2i}=x_{2i+2}$. This gives $x_{2i}=x_2^i$. Hence $H^*(\cproj,\Z)\simeq \Z[x_2]$. +\end{example} + +\begin{example} +We will compute the cup product structure of $H^*(\Omega S^n;\Z)$ using the path space fibration of $S^n$ for $n\geq2$. The additive structure is the same as for homology, and we can name the generators as in the figure, where $a_0=1$. +\begin{center} +\begin{tikzpicture}[>=stealth',auto,node distance=1.5cm, + main node/.style={font=\sffamily\bfseries},text height=1.5ex] + \node[main node] (00) at (0,0) {$\Z a_0$}; + \node[main node] (10) at (1.5,0) {$\Z a_0x$}; + \node[main node] (01) at (0,1) {$\Z a_1$}; + \node[main node] (11) at (1.5,1) {$\Z a_1x$}; + \node[main node] (02) at (0,2) {$\Z a_2$}; + \node[main node] (12) at (1.5,2) {$\Z a_2x$}; + \node[main node] (03) at (0,3) {$\Z a_3$}; + \node[main node] (13) at (1.5,3) {$\Z a_3x$}; + \node[main node] (04) at (0,4) {$\vdots$}; + \node[main node] (14) at (1.5,4) {$\vdots$}; + \node at (-1.4,0) {0}; + \node at (-1.4,1) {$n-1$}; + \node at (-1.4,2) {$2n-2$}; + \node at (-1.4,3) {$3n-3$}; + \node at (0,-1) {0}; + \node at (1.5,-1) {$n$}; + \node at (1.8,-0.7) {$p$}; + \node at (-1,4.5) {$q$}; + \draw (-0.7,-0.5) -- (2,-0.5); + \draw (-0.7,-0.5) -- (-0.7,5); + \path[every node/.style={font=\sffamily\small}] + (01) edge [->] (10) + (02) edge [->] (11) + (03) edge [->] (12); +\end{tikzpicture} +\end{center} +We may assume that $d(a_{k+1})=a_kx$ and note that $a_kx=xa_k$. + +We distinguish two cases. + +\emph{If $n$ is odd} we compute by induction to $i+j$ that $a_ia_j={i+j \choose i}a_{i+j}$. Hence $H^*(\Omega S^n,\Z)\simeq \Gamma_{\Z}[a_1]$, where the \emph{divided polynomial algebra} $\Gamma_R[\alpha]$ is the quotient of the free $R$-algebra $R[\alpha_1,\alpha_2,\ldots]$ by the relations $\alpha_i\alpha_j={i+j \choose i}\alpha_{i+j}$. + +\emph{If $n$ is even}, then we compute $a_1^2=0$ and by induction on $k$ we compute $a_1a_{2k}=a_{2k+1}$ and $a_1a_{2k+1}=0$ and $a_2^k=k!a_{2k}$. + + +Now $H^*(\Omega S^n,\Z)\simeq \Lambda_\Z[a_1]\otimes \Gamma_\Z[a_2]$ where the \emph{exterior algebra} $\Lambda_R[\alpha_1,\alpha_2,\ldots]$ is the free $R$-module with basis finite products $\alpha_{i_1}\cdots\alpha_{i_k}$ for $i_1<\cdots3$. Now convert the map $F\to S^3$ into a fibration. By the LES we see that the fiber is $K(\Z,2)=\cproj$. + +\begin{center} +\begin{tikzpicture}[auto,node distance=1.5cm, + main node/.style={font=\sffamily\bfseries},text height=1.5ex] + \node[main node] (S3) at (0,0) {$S^3$}; + \node[main node] (Z) [above of=S3] {$Z$}; + \node[main node] (K3) [right of=Z] {$K(\Z,3)$}; + \node[main node] (F) [left of=Z] {$F$}; + \node[main node] (X) [left of=S3] {$X$}; + \node (K2) [left of=X] {$\cproj$}; + \path[every node/.style={font=\sffamily\small}] + (F) edge [->] (Z) + (F) edge [->] (S3) + (F) edge [->] node [right] {$\wr$} (X) + (S3) edge [->] node [right] {$\wr$} (Z) + (Z) edge [->>] (K3) + (K2) edge [->] (X) + (X) edge [->>] (S3) + (S3) edge [->] node [below right] {$f$} (K3); +\end{tikzpicture} +\end{center} + +We now use the Serre Spectral Sequence of this last fibration. We know the homology groups of $S^3$ and $\cproj$, so we know the second page looks like this. Here the arrows are \emph{not} all isomorphisms. + +\begin{center} +\begin{tikzpicture}[>=stealth',auto,node distance=1.5cm, + text height=1.5ex] + \node (00) at (0,0) {$\Z 1$}; + \node (10) at (1.5,0) {$\Z x$}; + \node (01) at (0,1) {$\Z a$}; + \node (11) at (1.5,1) {$\Z ax$}; + \node (02) at (0,2) {$\Z a^2$}; + \node (12) at (1.5,2) {$\Z a^2x$}; + \node (03) at (0,3) {$\Z a^3$}; + \node (13) at (1.5,3) {$\Z a^3x$}; + \node (04) at (0,4) {$\vdots$}; + \node (14) at (1.5,4) {$\vdots$}; + \node at (-1.4,0) {0}; + \node at (-1.4,1) {$2$}; + \node at (-1.4,2) {$4$}; + \node at (-1.4,3) {$6$}; + \node at (0,-1) {0}; + \node at (1.5,-1) {$3$}; + \node at (1.8,-0.7) {$p$}; + \node at (-1,4.5) {$q$}; + \draw (-0.7,-0.5) -- (2,-0.5); + \draw (-0.7,-0.5) -- (-0.7,5); + \path[every node/.style={font=\sffamily\small}] + (01) edge [->] (10) + (02) edge [->] (11) + (03) edge [->] (12); +\end{tikzpicture} +\end{center} +\end{example} + +Since $X$ is 3-connected, $d:\Z a \to \Z x$ must be an iso, so we may assume $da=x$. Then $d(a^n)=na^{n-1}x$. Now we know what groups survive until $E_\infty$, to compute +\begin{align*} +H^i(X;\Z)&=\begin{cases} \Z_n & \text{if $i=2n+1$} \\ 0 & \text{if $i=2n$,} \end{cases} && \text{hence}& +H_i(X;\Z)&=\begin{cases} \Z_n & \text{if $i=2n>0$} \\ 0 & \text{if $i=2n-1$.} \end{cases} +\end{align*} +The Hurewicz Theorem modulo $\mc C$ now implies that the first $p$-torsion in $\pi_*(X)$, hence also in $\pi_*(S^3)$ is a $\Z_p$ in $\pi_{2p}$. For $p=2$ we get the stronger result that $\pi_4(S^3)=\Z^2$, hence also $\pi_{n+1}(S^n)=\Z_2$ for $n\geq3$. + +\end{document}