Remark

One calls the Poincaré polynomial of M relative to $\varphi$ the element $P_M(t) = \sum \varphi([M_n]) t^n \in G \otimes \mathbf{Z}[t, t^{-1}]$. One has $P_M(1) = \varphi([M])$ and $P_M(-1) = \chi(M)$.