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I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that classifying lens spaces was I problem that motivated Reidemeister to introduce torsion.

All of the definitions of torsion that I have seen are totally opaque to me. How do people think of the torsion of a chain complex and how in trying to classify lens spaces could I have been led to defining/computing torsions?

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    $\begingroup$ I recall Turaev works out the details of lens spaces in his book Torsions of 3-Dimensional Manifolds beginning with a cell decomposition and accompanying explicit bases for homology. It was very clarifying (but not, I fear, so clarifying that I remember enough to write an answer off the top of my head). $\endgroup$
    – Neal
    Nov 14, 2017 at 11:13

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Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$

$$0\to U_0 \stackrel{\pa}{\to} U_1\to 0.$$

(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)

This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.

Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.

The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their respective ambient spaces. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.)

A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also an $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,\ue_1)$.

Observe that if $\uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp $L_1$, then

$$|\det M(\pa,\ue_0,\ue_1)|=|\det M(\pa,\uf_0,\uf_1)|=|L_1/\pa L_0|, $$

where $|S|$ denotes the cardinality of the set $S$.

We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined (up to a sign) to be the number

$$\tau(\pa,L_0,L_1):=\pm \frac{1}{\det M(\pa,\ue_0,\ue_1)} =\pm \frac{1}{|L_1/\pa L_0|}. $$

The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology with local coefficients, or the chain complex associated to a $CW$-decomposition. You get one such invariant for every triangulation and every choice of local coefficients. For the Reidemeister torsion, the local coefficients are Abelian and correspond to group morphisms $\pi_1\to \mathbb{C}^*$.

It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.

This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.

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