Whitehead torsion

Definition

Let $R$ be a ring and let $C_*$ be a chain complex of $R$-modules with only finitely many nonzero terms. Each term is a finitely generated free module with a chosen basis. Assume $C_*$ is acyclic, and choose a chain contraction $\delta$, that is, maps $\delta \colon C_i \to C_{i+1}$ with $d \delta + \delta d = \mathrm{id}$. The map $d + \delta \colon \bigoplus_i C_{2i+1} \to \bigoplus_i C_{2i}$ is an isomorphism, and the torsion $\tau(C_*)$ is the class of its matrix in $\tilde K_1(R) = K_1(R)/\set{\pm 1}$, using K1 of a ring.

Now let $f \colon K \to L$ be a cellular homotopy equivalence of finite connected CW complexes. Let $\pi$ be the fundamental group of $L$, and let $M_f$ be the mapping cylinder of $f$. Pass to universal covers. The relative cellular complex $C_*(\tilde M_f, \tilde K)$ is an acyclic complex of finitely generated free modules over the group ring $\Z[\pi]$. It has a basis given by lifts of the cells of $M_f$ not in $K$. The Whitehead torsion $\tau(f)$ is the image of $\tau(C_*(\tilde M_f, \tilde K))$ in the Whitehead group $\mathrm{Wh}(\pi)$.