Poincaré Duality Group

An integral Poincaré duality group of dimension is a group for which the trivial -module has a finite resolution by finitely generated projective modules, and the group cohomology with group-ring coefficients satisfies

The right -action on the last group records the orientation character. The group is oriented when this action is trivial.

The fundamental group of a closed aspherical manifold satisfies this duality condition. For example, is an oriented integral Poincaré duality group of dimension . The converse manifold-realization question requires finiteness assumptions beyond duality.