Intuition for torsion of a chain complex and application to lens spaces 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? 
 A: 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.
