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 $\chi(X)=\sum_{i \geq0}(-1)^i \#(i\text{-cells of } X)\in\Z$.
\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
and note that we have the exact sequences $0\to\Im\partial\to C_0\to\underbrace{H_0}_{= C_0/\Im\partial}\to0$ and $0\to\underbrace{H_1}_{=\Ker\partial}\to C_1\xrightarrow{\partial}\Im\partial\to0$.
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
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
