= 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 $\mathbb ZH$ coincide and give $\mathbb ZG$, 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 $PD^1$ subgroup $\mathbb Z$ but has infinite integral cohomological dimension.
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