In their book Lecture Notes in Algebraic Topology, Davis and Kirk define the torsion of an acyclic chain complex $C$ in the following way:

Since $C$ is acyclic, there exists a simple chain complexes $E,F$ and a chain isomorphism $f: E \rightarrow F \bigoplus C$ (a simple chain complex is a finite direct sum of elementary chain complexes which have only two non-zero terms and the identity map between these two terms). Then define the torsion $\tau(C)$ of $C$ as $\sum_i (-1)^n [f_n: E_n \rightarrow F_n \bigoplus C_n] \in \tilde{K_1}(R)$, which can be shown to be independent of $E,F$. There is a copy of the book at http://www.indiana.edu/~lniat/book.pdf from Kirk's website which contains this material in Chapter 11.

Another definition of the torsion is as follows: Let $C$ be an acyclic chain complex with $C_n$ a free $R$ module with specified basis. Then there exists a chain homotopy $s$ (which is a sequence of maps $s_n: C_n \rightarrow C_{n+1}$) between the identity and zero map. Then $d + s$ is an isomorphism between the $R$-modules $C_{odd}$ and $C_{even}$. Then we can consider $[d+s] \in \tilde{K_1}(R)$. This is the same as the wikipedia definition of Whitehead torsion at http://en.wikipedia.org/wiki/Whitehead_torsion.

I was wondering if someone can help me see why these two definitions are the same. Also, Exercise 198 of the Davis/Kirk book is to show that these two definitions are the same. I was able to use $s$ to construct explicit $E,F$ and a map $f$ between $E$ and $F \bigoplus C$ but I did not get equivalence of the two definitions. I got the following modules for $E$ and $F$:

$E_n = C_1 \bigoplus C_2 \bigoplus C_3 \bigoplus ... \bigoplus C_n$

$F_n = C_1 \bigoplus C_2 \bigoplus C_3 \bigoplus ... \bigoplus C_{n-1}$.


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