From fd17083d8225985e2f8b61464f3381865b7d5254 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Thu, 2 Oct 2014 17:50:12 +0200 Subject: [PATCH] Adds some basic stuff and some Serre stuff --- thesis/notes/Basics.tex | 89 +++++++++++++++++++++++++++++++++++++++++ thesis/notes/Serre.tex | 39 ++++++++++++++++++ thesis/preamble.tex | 20 ++++++++- thesis/thesis.tex | 2 + 4 files changed, 149 insertions(+), 1 deletion(-) create mode 100644 thesis/notes/Basics.tex create mode 100644 thesis/notes/Serre.tex diff --git a/thesis/notes/Basics.tex b/thesis/notes/Basics.tex new file mode 100644 index 0000000..2b6b859 --- /dev/null +++ b/thesis/notes/Basics.tex @@ -0,0 +1,89 @@ + +\section{Rational homotopy theory} +\label{sec:rational} + +In this section we will state the aim of rational homotopy theory. Moreover we will recall classical theorems from algebraic topology and deduce rational versions of them. + +In the following definition \emph{space} is to be understood as a topological space or a simplicial set. We will restrict to simply connected spaces. + +\Definition{rational-space}{ + A space $X$ is a \emph{rational space} if + $$ \pi_i(X) \text{ is a $\Q$-vectorspace } \quad\forall i > 0. $$ +} + +\Definition{rational-homotopy-groups}{ + We define the \emph{rational homotopy groups} of a space $X$ as: + $$ \pi_i(X) \tensor \Q \quad \forall i > 0.$$ +} + +Note that for a rational space $X$, the homotopy groups are isomorphic to the rational homotopy groups, i.e. $\pi_i(X) \tensor \Q \iso \pi_i(X)$. + +\Definition{rational-homotopy-equivalence}{ + A map $f: X \to Y$ is a \emph{rational homotopy equivalence} if $\pi_i(f) \tensor \Q$ is a linear isomorphism for all $i > 0$. +} + +\Definition{rationalization}{ + A map $f: X \to X_0$ is a \emph{rationalization} if $X_0$ is rational and $f$ is a rational homotopy equivalence. +} + +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 iff $f$ is a weak equivalence. + +We will later see that any space admits a rationalization. The theory of rational homotopy theory is then the study of the homotopy category $\Ho_\Q(\Top) \iso \Ho(\Top_\Q)$, which is on its own turn equivalent to $\Ho(\sSet_\Q) \iso \Ho_\Q(\sSet)$. + +\subsection{Classical results from algebraic topology} + +We will now recall known results from algebraic topology, without proof. One can find many of these results in basic text books, such as [May, Dold, ...]. Note that all spaces are assumed to be $1$-connected. + +\Theorem{relative-hurewicz}{ + (Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map + $$ h_i : \pi_i(X, A) \to H_i(X, A). $$ + If furhtermore $(X,A)$ is $n$-connected, then the map $h_i$ is an isomorphism for all $i \leq n + 1$ +} + +\Theorem{serre-les}{ + (Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence: + $$ \cdots \tot{\del} \pi_i(F) \tot{i_\ast} \pi_i(X) \tot{f_\ast} \pi_i(Y) \tot{\del} \cdots \to \pi_0(Y) \to \ast, $$ + where $F$ is the fibre of $f$. +} + +Using an inductive argument and the previous two theorems, one can show the following theorem (as for example shown in \cite{griffith}). +\Theorem{whitehead-homology}{ + (Whitehead) For any map $f: X \to Y$ we have + $$ \pi_i(f) \text{ is an isomorphism } \forall 0 < i < r \iff H_i(f) \text{ is an isomorphism } \forall 0 < i < r. $$ + In particular we see that $f$ is a weak equivalence iff it induces an isomorphism on homology. +} + +The following two theorems can be found in textbooks about homological algebra, such as [Weibel]. +\Theorem{universal-coefficient}{ + (Universal Coefficient Theorem) + For any space $X$ and abelian group $A$, there are natural short exact sequcenes + $$ 0 \to H_n(X) \tensor A \to H_n(X; A) \to \Tor(H_{n-1}(X), A) \to 0, $$ + $$ 0 \to \Ext(H_{n-1}(X), A) \to H^n(X; A) \to \Hom(H_n(X), A) \to 0. $$ +} + +\Theorem{kunneth}{ + (Künneth Theorem) + For spaces $X$ and $Y$, there is a short exact sequence + $$ 0 \to H(X; A) \tensor H(Y; A) \to H(X \times Y; A) \to \Tor(H(X; A), H(Y; A)) \to 0, $$ + where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. +} + +\subsection{Immediate results for rational homotopy theory} + +The latter two theorems have a direct consequence for rational homotopy theory. By taking $A = \Q$ we see that the torsion groups vanish. We have the immediate corollary. + +\Corollary{rational-corollaries}{ + We have the following natural isomorphisms + $$ H(X) \tensor \Q \tot{\iso} H(X; \Q), $$ + $$ H^n(X; \Q) \tot{\iso} \Hom(H(X); \Q), $$ + $$ H(X \times Y) \tot{\iso} H(X) \tensor H(Y). $$ +} + +The long exact sequence for a Serre fibration also has a direct consequence for rational homotopy theory. +\Corollary{rational-les}{ + Let $f: X \to Y$ be a Serre fibration, then there is a natural long exact sequence of rational homotopy groups: + $$ \cdots \tot{\del} \pi_i(F) \tensor \Q \tot{i_\ast} \pi_i(X) \tensor \Q \tot{f_\ast} \pi_i(Y) \tensor \Q \tot{\del} \cdots, $$ +} + +In the next sections we will prove the rational Hurewicz and rational Whitehead theorems. These theorems are due to Serre [Serre]. + diff --git a/thesis/notes/Serre.tex b/thesis/notes/Serre.tex new file mode 100644 index 0000000..c18e420 --- /dev/null +++ b/thesis/notes/Serre.tex @@ -0,0 +1,39 @@ + +\section{Serre theorems mod $C$} + +In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in [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. + +\Lemma{whitehead-decomposition}{ + (Whitehead Decomposition) + For a space X, we have a decomposition in fibrations: + $$ \cdots \fib X(n+1) \fib X(n) \fib X(n) \fib \cdots \fib X(1) = X, $$ + such that: + \begin{itemize} + \item $K(\pi_n(X), n-1) \cof X(n+1) \fib X(n)$ is a fiber sequence, + \item There is a space $X'_n$ weakly equivelent to $X(n)$ such that $X(n+1) \ cof X'_n \fib K(\pi_n(X), n)$ is a fiber sequence, and + \item $\pi_i(X(n)) = 0$ for all $i < n$ and $\pi_i(X(n)) \iso \pi_i(X)$ for all $i \leq n$. + \end{itemize} +} + +\Theorem{absolute-serre-hurewicz}{ + (Absolute Serre-Hurewicz Theorem) + Let $C$ be a Serre-class of abelian groups. Let $X$ a $1$-connected space. + If $\pi_i(X) \in C$ for all $i