Equivariant Poincaré–Alexander–Lefschetz duality and the Cohen–Macaulay property

Abstract

We prove a Poincaré–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.