Abstract

Via Poincaré duality, fundamental groups of aspherical manifolds have (appropriately shifted) isomorphisms between their homology and cohomology. In a 1973 Inventiones paper, Bieri and Eckmann defined a broader notion of a duality group, where the isomorphism between homology and cohomology can be twisted by an object called a dualizing module. Examples of these groups in geometric group theory (after passing to a finite-index subgroup) include $GL(n,/mathbb{Z})$, mapping class groups, and automorphism groups of free groups.

Every example of a duality group that we know of has a classifying space satisfying a local condition called the Cohen-Macaulay property. Such spaces also satisfy weaker (twisted) versions of Poincaré duality via their local homology sheaves (or local cohomology cosheaves). However it is not clear on how to directly link the topological and algebraic duality results. The goal of the talk is to explain more about the words used in the above paragraphs and say where we have got to so far with this problem. Based on joint work with Thomas Wasserman.