From 8dc3dbb161742681f0ff20152b9cc667fc74cc5b Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Mon, 22 Dec 2014 15:55:32 +0100 Subject: [PATCH] Adds details to the Serre section --- thesis/Makefile | 8 ++ .../Applications_And_Further_Topics.tex | 2 +- thesis/diagrams/Kreck_Exact_Sequence.tex | 3 +- thesis/notes/Basics.tex | 2 +- thesis/notes/Serre.tex | 90 +++++++++++-------- thesis/references.bib | 11 +++ 6 files changed, 76 insertions(+), 40 deletions(-) diff --git a/thesis/Makefile b/thesis/Makefile index 652de5e..1933501 100644 --- a/thesis/Makefile +++ b/thesis/Makefile @@ -12,7 +12,15 @@ thesis: dirs cp build/thesis.pdf ./ fast: dirs + cp references.bib build/ xelatex -file-line-error -output-directory=build thesis.tex + cd build; bibtex thesis + cp build/thesis.pdf ./ + +haltfast: dirs + cp references.bib build/ + xelatex -file-line-error -output-directory=build --halt-on-error thesis.tex + cd build; bibtex thesis cp build/thesis.pdf ./ images: dirs diff --git a/thesis/chapters/Applications_And_Further_Topics.tex b/thesis/chapters/Applications_And_Further_Topics.tex index 45f6279..cd63591 100644 --- a/thesis/chapters/Applications_And_Further_Topics.tex +++ b/thesis/chapters/Applications_And_Further_Topics.tex @@ -94,7 +94,7 @@ $$ |K(A \tensor B)| \iso |K(A)| \times |K(B)|. $$ In this section we will prove that the rational cohomology of an H-space is free as commutative graded algebra. We will also give its minimal model and relate it to the homotopy groups. In some sense H-spaces are homotopy generalizations of topological monoids. In particular topological groups (and hence Lie groups) are H-spaces. \Definition{H-space}{ - An H-space is a pointed topological space $x_0 \in X$ with a map $\mu: X \times X \to X$, such that $\mu(x_0, -), \mu(-, x_0) : X \to X$ are homotopic to $\id_X$. + An \Def{H-space} is a pointed topological space $x_0 \in X$ with a map $\mu: X \times X \to X$, such that $\mu(x_0, -), \mu(-, x_0) : X \to X$ are homotopic to $\id_X$. } Let $X$ be an H-space, then we have the induced map $\mu^\ast: H^\ast(X; \Q) \to H^\ast(X; \Q) \tensor H^\ast(X; \Q)$ on cohomology. Because homotopic maps are sent to equal maps in cohomology, we get $H^\ast(\mu(x_0, -)) = \id_{H^\ast(X; \Q)}$. Now write $H^\ast(\mu(x_0, -)) = (\counit \tensor \id) \circ H^\ast(\mu)$, where $\counit$ is the augmentation induced by $x_0$, to conclude that for any $h \in H^{+}(X; \Q)$ the image is of the form diff --git a/thesis/diagrams/Kreck_Exact_Sequence.tex b/thesis/diagrams/Kreck_Exact_Sequence.tex index 6d28d81..2c7aff7 100644 --- a/thesis/diagrams/Kreck_Exact_Sequence.tex +++ b/thesis/diagrams/Kreck_Exact_Sequence.tex @@ -1,4 +1,5 @@ +\small \xymatrix{ - \cdots \ar[r] & H_{i+1}(E^{k+1}, E^k) \ar[r] \arwe[d] & H_i(E^k, F) \ar[r] \ar[d] & H_i(E^{k+1}, F) \ar[r] \ar[d] & \cdots \\ + \cdots \ar[r] & H_{i+1}(E^{k+1}, E^k) \ar[r] \ar[d]^\iso & H_i(E^k, F) \ar[r] \ar[d] & H_i(E^{k+1}, F) \ar[r] \ar[d] & \cdots \\ \cdots \ar[r] & H_{i+1}(B^{k+1}, B^k) \ar[r] & H_i(B^k, b_0) \ar[r] & H_i(B^{k+1}, b_0) \ar[r] & \cdots } diff --git a/thesis/notes/Basics.tex b/thesis/notes/Basics.tex index d59b9fd..192c6ce 100644 --- a/thesis/notes/Basics.tex +++ b/thesis/notes/Basics.tex @@ -30,7 +30,7 @@ Note that for a rational space $X$, the ordinary homotopy groups are isomorphic Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence if and only if $f$ is a weak equivalence. -The theory of rational homotopy theory is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{Quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1) = \Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences). +The theory of rational homotopy theory is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1) = \Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences). \section{Classical results from algebraic topology} diff --git a/thesis/notes/Serre.tex b/thesis/notes/Serre.tex index fc2c474..a6da100 100644 --- a/thesis/notes/Serre.tex +++ b/thesis/notes/Serre.tex @@ -1,14 +1,15 @@ \chapter{Serre theorems mod \texorpdfstring{$\C$}{C}} -\label{sec:serre} In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in \cite{serre}. In his paper he considered homology groups `modulo a class of abelian groups'. In our case of rational homotopy theory, this class will be the class of torsion groups. \Definition{serre-class}{ A class $\C \subset \Ab$ is a \Def{Serre class} if \begin{itemize} - \item for all exact sequences $0 \to A \to B \to C \to 0$ with $A$ and $C$ in $\C$, $B$ also belongs to $\C$, - \item $\C$ is closed under taking direct sums (both finite and infinite). + \item for all exact sequences $0 \to A \to B \to C \to 0$ if two abelian groups are in $\C$, then so is the third, + \item for all $A \in \C$ the tensor product $A \tensor B$ is in $\C$ for any abelian group $B$, + \item for all $A \in \C$ the Tor group $\Tor(A, B)$ is in $\C$ for any abelian group $B$, and + \item for all $A \in \C$ the group-homology $H_i(A; \Z)$ is in $\C$ for all positive $i$. \end{itemize} } @@ -23,37 +24,18 @@ Serre gave weaker axioms for his classes and proves some of the following lemmas \end{itemize} } -The most important properties we need of a Serre class are the following: -\Lemma{Serre-properties}{ - Let $\C \subset \Ab$ be a Serre class. Then we have: - \begin{enumerate} - \item If $A \in \C$, then $A \tensor B \in \C$ for all $B$. - \item If $A \in \C$, then $H(A) \in \C$ ($H(A)$ is the group homology of $A$). - \end{enumerate} -} +As noted by Hilton in \cite{hilton} we think of Serre classes as a generalized 0. This means that we can also express some kind of generalized injective and surjectivity. Here we only need the notion of a $\C$-isomorphism: \Definition{serre-class-maps}{ - Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a $\C$-isomorphism if both the kernel and cokernel lie in $\C$. + Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a \Def{$\C$-isomorphism} if both the kernel and cokernel lie in $\C$. } -Note that the map $0 \to C$ is a $\C$-isomorphism for any $C \in \C$. \todo{Er missen nog wat eigenschappen voor tensors} - -\Lemma{serre-class-rational-iso}{ - Let $\C$ be the Serre class of all torsion groups. Then - $f$ is a $\C$-iso $\iff$ $f \tensor \Q$ is an isomorphism. -} -\Proof{ - First note that if $C \in \C$ then $C \tensor \Q = 0$. - - Then consider the exact sequence - $$ 0 \to \ker(f) \to A \tot{f} B \to \coker(f) \to 0 $$ - and tensor this sequence with $\Q$. In this tensored sequence the kernel and cokernel vanish if and only if $f \tensor \Q$ is an isomorphism. -} +Note that the maps $0 \to C$ and $C \to 0$ are $\C$-isomorphisms for any $C \in \C$. More importantly the 5-lemma also holds for $\C$-isos and whenever $f$, $g$ and $g \circ f$ are maps such that two of them are $\C$-iso, then so is the third. -In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences, \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more elementary proof using cellular homology. +In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences, \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more geometric proof. \Lemma{kreck}{ - Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $1$-connected spaces and $F$ its fiber. If $\RH_i(F) \in \C$ for all $i < n$, then + Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $0$-connected spaces and $F$ its fiber. If $\RH_i(F) \in \C$ for all $i < n$, then \begin{itemize} \item $H_i(E, F) \to H_i(B, b_0)$ is a $\C$-iso for $i \leq n+1$ and \item $H_i(E) \to H_i(B)$ is a $\C$-iso for all $i \leq n$. @@ -95,17 +77,18 @@ In the following arguments we will consider fibrations and need to compute homol } \Lemma{homology-em-space}{ - Let $\C$ be a Serre class and $C \in \C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(C, n)) \in \C$. + Let $\C$ be a Serre class and $G \in \C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(G, n)) \in \C$. } \Proof{ - We prove this by induction on $n$. The base case $n = 1$ follows from group homology. + We prove this by induction on $n$. The base case $n = 1$ follows from group homology as the construction of $K(G, 1)$ can be used to obtain a projective resolution of $\Z$ as $\Z[G]$-module \todo{reference}. This then identifies the homology of the Eilenberg-MacLane space with the group homology, we get for $i>0$ an isomorphism + $$ H_i(K(G, 1); \Z) \iso H_i(G; \Z) \in \C. $$ - For the induction we can use the loop space and \LemmaRef{kreck}. - - \todo{Bewijs afmaken} + Suppose we have proven the statment for $n$. If we consider the case of $n+1$ we can use the path fibration to relate it to the case of $n$: + $$ \Omega K(G,n+1) \to P K(G, n+1) \fib K(G, n+1) $$ + Now $\Omega K(G, n+1) = K(G, n)$, and we can apply \LemmaRef{kreck} as the reduced homology of the fiber is in $\C$ by induction hypothesis. Conclude that the homology of $P K(G, n+1)$ is $\C$-isomorphic to the homology of $K(G, n)$. Since $\RH_\ast(P K(G, n+1)) = 0$, we get $\RH_\ast(K(G, n+1)) \in \C$. } -For the main theorem we need the following construction. \todo{Geef de constructie} +For the main theorem we need the following construction. \todo{Geef de constructie of referentie} \Lemma{whitehead-tower}{ (Whitehead tower) We can decompose a $0$-connected space $X$ into fibrations: @@ -124,7 +107,9 @@ For the main theorem we need the following construction. \todo{Geef de construct If $\pi_i(X) \in C$ for all $i