 Finite-index invariance of Poincare duality for torsion-free groups

For a torsion-free group $G$ and finite-index subgroup $H$, $G$ is a Poincare duality group of dimension $n$ if and only if $H$ is. Finite-index induction and coinduction of $\mathbb{Z}H$ coincide and give $\mathbb{Z}G$, so Shapiro's lemma identifies their group-ring cohomology as abelian groups. The finite-index finiteness and cohomological-dimension lemmas supply finite projective resolutions. Without torsion-freeness of $G$ the converse fails: $\mathbb{Z}\times C_2$ contains the index-two $P{D}^1$ subgroup $\mathbb{Z}$ but has infinite integral cohomological dimension.