Commit ac43f39b authored by Jan-Bernhard Kordaß's avatar Jan-Bernhard Kordaß

Added first lecture draft.

parent e4c548b4
% !TEX root = skript-group-actions-in-riemannian-geometry.tex
\tableofcontents
\section*{}
\chapter{Tosion Invariants}
Torsion invariants fall into a class of so-called ``secondary invariants'' of topological spaces in the sense that they are only defined if a certain class of ``primary invariants'' (e.g. Betti numbers) vanish.
Often they reveal more subtle geometric information.
The following will contain a discussion of Whitehead and Reidemeister torsion.
Informally, corresponding primary invariants are Lefschetz numbers (Whitehead torsion) and the Euler characteristic (Reidemeister torsion).
\section{Review of Euler characteristic and Lefschetz numbers.}
\subsection{CW Complexes}
\begin{dfn*}
A (finite) \CmMark{CW-complex} is a hausdorff space with a decomposition $E$ into (finitely many) cells (space hemeomorphic to some $\R^n$) such that for every $e \in E$ there is a continuous map $\phi_e \colon D^n \to X$ with $\phi_e \colon \mathring D^n \xrightarrow{\cong} e$ and $\Im(\phi_e|_{S^{n-1}}) \subset \bigcup_{f \in E, \dim f \leq n-1} f$.
\end{dfn*}
\begin{expl*}
\begin{enumerate}
\item Simplicial complexes, e.g. triangles, pyramides, etc.
\item But CW-complexes are more general, the following graph is CW for example:
\begin{center}
\begin{tikzpicture}
\draw (0,0) to[bend left] (2,0);
\draw (0,0) to[bend right] (2,0);
\draw (2,0) to (3,0);
\end{tikzpicture}
\end{center}
One can even attach a disc along its boundary to a single 1-cell.
\end{enumerate}
\end{expl*}
\subsection{Euler characteristic}
\begin{dfn*}
The Euler class $\chi(X)$ of a finite CW-complex $X$ is defined as $\chi(X) = \sum_{i \geq 0}(-1)^i \#(i\text{-cells of } X) \in \Z$.
\end{dfn*}
\begin{thm*}[Euler-Poincaré]
\begin{align*}
\chi(X) = \sum_{i \geq 0} (-1)^i b_i(X),
\end{align*}
where $b_i(X) = \rk_{\Z} H_i(X;\Z)$.
\end{thm*}
In particular, $\chi$ is a homotopy invariant.
\begin{proof}[``Proof'']
$H_i(X;\Z) = H_i(C_{*}^{CW}(X))$, where $C_{*}^{CW}(X)$ is the cellular chain complex
\begin{align*}
\cdot \to C_{i+1}^{CW}(X)\xrightarrow{\partial} \underbrace{C_{i}^{CW}(X)}_{\cong \Z^{\# i\text{-cells}}} \xrightarrow{\partial} C_{i-1}^{CW}(X) \to \cdots
\end{align*}
Thus $\chi(C_{*}) := \sum_{i \geq 0} (-1)^i\rk_{\Z}(C_{i})$ and $\chi(C^{CW}(X)) = \chi(X)$.
This boils down to
\begin{align*}
\chi(C_{*}) = \sum_{i \geq 0} \rk_{\Z}H_i(C_{*}) ( = \chi(H_{*}(C_{*}))].
\end{align*}
This is just additivity of the rank!
Consider
\begin{align*}
C_1 \xrightarrow{\partial} C_0
\end{align*}
and note that we have the exact sequences $0 \to \Im \partial \to C_0 \to \underbrace{H_0}_{= C_0/\Im \partial} \to 0$ and $0 \to \underbrace{H_1}_{= \Ker \partial} \to C_1 \xrightarrow{\partial} \Im \partial \to 0$.
Thus $\chi(C_{*)} = \rk_{\Z} C_0 - \rk_{\Z} C_1 = \rk_{\Z} \Im \partial + \rk_{\Z} H_0 - \rk_{\Z}H_1 - \rk_{\Z} \Im \partial = \rk H_0 - \rk H_1$, which completes the ``proof''.
\end{proof}
\subsection{Review of cellular homology}
Let $X$ be a CW-complex with cellular decomposition $E$.
Then we can consider the \CmMark{n-skeleton}
\begin{align*}
X^n := \sum_{e \in E, \dim e \leq n} e,
\end{align*}
which yields a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such there is a pushout diagram
\begin{equation*}
\begin{tikzcd}
\coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\
\coprod D^n \ar{r} & X^n
\end{tikzcd}
\end{equation*}
One could take this as an alternative definition of a CW-complex by a filtration with the pushout property.
The cells can be recovered as connected components of $X^n\setminus X^{n-1}$.
We have
\begin{align*}
C_i^{CW}(X) = H_i(X^i, X^{i+1}) \xleftarrow{\cong} H_i(\coprod D^i, \coprod S^{i-1}) \cong \bigoplus H_i(D^i, S^{i-1}) \cong \bigoplus \Z^{\# i\text{-cells}},
\end{align*}
where the first isomorphism $\leftarrow$ is given by excision.
The boundary maps $C_i^{CW}(X) \xrightarrow{\partial} C_{i-1}^{CW}(X) $ come from
\begin{align*}
H_i(X^i,X^{i-1}) \to H_{i-1}(X^{i-1}) \to H_{i-1}(X^{i-1},X^{i-2}).
\end{align*}
Under this isomorphism, the matrix entry belonging to $(e,f)$ where $e$ is an $n$-cell, $f$ an $(n-1)$-cell is the \CmMark{degree} of the map.
\begin{align*}
S^{i-1} \xrightarrow{\phi_e|_{S^{n-1}}} X^{i-1} \xrightarrow{\operatorname{proj}} X^{i-1}/(X^{i-1}\setminus f) \xleftarrow{\phi_f, \cong} D^{i-1}/S^{i-2} \cong S^{i-1}.
\end{align*}
\begin{expl*}
Consider the torus as an identification square.
We convince ourselves that the cellular chain complex is given as $\Z \to \Z \oplus \Z \to \Z$, where $1 \mapsto (0, 0)$, since it is described by a map $S^1 \to S^1$ traversing the 2-cell according to orientation has degree $0$.
\end{expl*}
\subsection{Lefschetz number}
Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, if $f(X^i) \subset Y^i$ for all $i$.
\begin{thm*}[Cellular approximation]
Any map between CW-complexes is homotopic to a cellular map.
\end{thm*}
\begin{dfn*}
The \CmMark{Lefschetz number} of a self-map $f \colon X \to X$ of a finite CW-complex is defnined as
\begin{align*}
\Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z.
\end{align*}
\end{dfn*}
\begin{rem*}
$\Lambda(\id_X) = \chi(X)$.
\end{rem*}
The following theorem yields a description of Lefschetz numbers by homology.
\begin{thm*}
$\Lambda(f) = \sum_{i \geq 0}(-1)^i \tr H_i(f)$.
\end{thm*}
Thus, this number only depends on the homotopy class of $f$.
\begin{proof}
Similar to the proof of Euler-Poincaré using the additivity of the trace, i.e. in the situation
\begin{equation}
\begin{tikzcd}[row sep=small]
0 \ar{r} & A \ar{r} \ar{d}{a} & B \ar{r} \ar{d}{b} & C \ar{r} \ar{d}{c} & 0\\
0 \ar{r} & A \ar{r} & B \ar{r} & C \ar{r} & 0
\end{tikzcd}
\end{equation}
we have $\tr(b) = \tr(a) + \tr(c)$.
\end{proof}
\begin{thm*}
If $f$ has no fixed point, then $\Lambda(f) = 0$.
\end{thm*}
\begin{rem*}
The converse is not true (think of counterexamples, e.g. $S^1 \wedge S^1$), although there is one in the case of simply-connected closed manifolds.
\end{rem*}
\begin{proof}
Let $X$ be metrizable and let $d$ be a metric.
If $X$ is compact, there exists an $\varepsilon > 0$ with $d(f(x), x) > 3\varepsilon$.
One can ``refine'' the CW-structure to a new one such that every cell has diameter $< \varepsilon$.
By cellular approximation we can see that there exists a cellular map $g \colon X \to X$ with $g \simeq f$ and $d(g(x),f(x)) < \varepsilon$.
Thus $g(\overline e) \cap \overline e = \emptyset$ for every cell $e$.
Hence, the diagonal matrix entries of each $C_i^{CW}(g)$ are zero and thus $\Lambda(g) = \Lambda(f) = 0$.
\end{proof}
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