Hey all,

I try to understand an argument in Lücks "A basic introduction to surgery theory" on page 51 which goes as follows:

Let $\mathbb{Z} \pi$ be the group ring where $\pi$ denotes the fundamental group of a finite connected CW-Komplex. Furthermore we regard the zellular $\mathbb{Z}\pi$-chain complex $C_*(\widetilde X) $ of the universal covering of $X$. Now it is claimed, that taking the n-th homology of the double complex $hom(C^{-*}(\widetilde X),C_*(\widetilde X))$ is the set of $\mathbb{Z}\pi$-chain homotopy classes of $\mathbb{Z}\pi$-chain maps from $C^{n-*}(\widetilde X)$ to $C_*(\widetilde X) $. Here $C^{n-*}(\widetilde X)$ denotes the dual chain complex given by $C^{n-*}(\widetilde X)_p:=hom(C_{n-p},\mathbb{Z}\pi) $ with the boundary map $ \circ \partial^C $.

If I regard the n-th boundary map of the double complex $hom(C^{-*}(\widetilde X),C_*(\widetilde X))$ I considered as definition of the n-th boundary map $\partial^n f_p :=f_p \circ\partial^{C^{-*}}+(-1)^{n-1}\partial^{C_*} \circ f_{p+1}$ ? This seems to be almost the definition of a chain homotopy but the prefactor $(-1)^{n-1} $ causes troubles.

I would be grateful for any references to literature.