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? 



Intuition for torsion of a chain complex and application to lens spaces
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](https://www3.nd.edu/%7Elnicolae/Torsion.pdf) where you will find many other descriptions and applications.


