Equivariant Poincar\'e-Alexander-Lefschetz duality and the Cohen-Macaulay property

TL;DR

This paper establishes a duality theorem for rational torus-equivariant cohomology and homology manifolds, extending classical results to non-compact and non-orientable spaces, and explores the Cohen-Macaulay property of related modules.

Contribution

It introduces a Poincaré-Alexander-Lefschetz duality for equivariant cohomology and homology, generalizing previous results to broader classes of spaces.

Findings

Proves duality theorem for non-compact, non-orientable spaces

Derives short exact sequences in equivariant cohomology from homology sequences

Highlights the role of Cohen-Macaulay modules in equivariant topology

Abstract

We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.