% !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} %%% Local Variables: %%% mode: latex ... ...