 Poincare duality group

A Poincare duality group of dimension $n$ has a finite-length resolution of the trivial group ring module $\mathbb{Z}$ by finitely generated projective modules, and $H^k(G,\mathbb{Z}G)$ vanishes except in degree $n$, where it is infinite cyclic as an abelian group. The top module $D$ can have an orientation-sign action. Reversing the dual projective resolution gives $H^k(G,V)\cong {\mathrm{Tor}}_{n-k}^{\mathbb{Z}G}(D,V)$, equivalently $H_{n-k}(G,D\otimes V)$ with the appropriate diagonal action. Untwisted duality requires trivial orientation action. In positive dimension $H^0(G,\mathbb{Z}G)=0$, since a finitely supported invariant element cannot exist for an infinite group.