Let $H\subset G$ be a subgroup. Let $E_*G$ be a free (right) $\mathbb ZG$-resolution of the trivial representation $\mathbb Z$. Because $E_*G$ is then also a free $\mathbb ZH$-resolution of the trivial representation, we can compute the homology of $H$ by $$H_i(H) = H_i(E_*G \otimes_{\mathbb ZH} \mathbb Z).$$ Let $g\in C_G(H)$ commute with all elements of $H$. Then there is a chain complex map automorphism of $E_*G\otimes_{\mathbb ZH} \mathbb Z$ given by $$ x \otimes 1 \longmapsto xg \otimes 1.$$
Prove that this map induces the identity on $H_i(H)$.
Notice that the map $x\mapsto xg$ is an endomorphism of the complex of $H$-modules $E_*G$, and it is in fact a lift of the identity of the right $H$-module $\mathbb Z$ to its resolution $E_*G$. It follows that that map of complexes is homotopic to the identity.