Bredon-Poincare Duality Groups

Abstract

If $G$ is a group which admits a manifold model for $\mathrm{B}G$ then $G$ is a Poincaré duality group. We study a generalisation of Poincaré duality groups, introduced initially by Davis and Leary, motivated by groups $G$ with cocompact manifold models $M$ for $\underline{\mathrm{E}}G$ where $M^H$ is a contractible submanifold for all finite subgroups $H$ of $G$. We give several sources of examples and constructions of these Bredon-Poincaré duality groups, including using the equivariant reflection group trick of Davis and Leary to construct examples of Bredon-Poincaré duality groups arising from actions on manifolds $M$ where the dimensions of the submanifolds $M^H$ are specified. We classify Bredon-Poincaré duality groups in low dimensions, and discuss behaviour under group extensions and graphs of groups.