Poincare duality group ($P{D}^n$)

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.

Let $N$ be a normal finitely presented group inside a dimension-three Poincare duality group $G$, with $G/N$ containing an element of infinite order. Then $N$ is either a free group or a dimension-two Poincare duality group. The Strebel infinite-index subgroup theorem gives $\mathrm{cd}N\le 2$. If the dimension is two, take the preimage of an infinite cyclic subgroup of the quotient. The cyclic-extension shift of group-ring cohomology gives dimension three, forcing this preimage to have finite index by Strebel. Its duality then gives $H^{\ast }(N;\mathbb{Z}N)$ concentrated in degree two with top group $\mathbb{Z}$. The quotient is virtually cyclic, and surface automorphism realization gives a finite-index surface mapping-torus group.

Every integral dimension-two Poincare duality group is the fundamental group of a closed aspherical topological surface. Orientable examples are the torus and closed surfaces of genus at least two. Nonorientable aspherical closed surfaces supply the orientation-twisted examples.

An infinite-index subgroup of an integral dimension-$d$ Poincare duality group has integral cohomological dimension at most $d-1$. No finiteness assumption on the subgroup is needed for this inequality. In dimension three it places every infinite-index subgroup in cohomological dimension at most two.

For an extension with kernel a dimension-$n$ Poincare duality group and quotient a dimension-$q$ Poincare duality group, apply the Lyndon–Hochschild–Serre spectral sequence to the ambient group ring. Kernel cohomology is concentrated in degree $n$ and is a regular quotient module with a rank-one orientation twist. Quotient cohomology is then concentrated in degree $q$ with underlying group $\mathbb{Z}$. The extension resolution supplies finiteness and gives cohomological dimension at most $n+q$, proving duality in that dimension. Orientation actions need not be trivial.

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.