Equivariant cohomology, syzygies and orbit structure

TL;DR

This paper links the exactness of the Atiyah-Bredon sequence in equivariant cohomology to syzygy conditions of the module, providing algebraic criteria for GKM method applicability in Poincare duality spaces.

Contribution

It establishes a new algebraic characterization of the Atiyah-Bredon sequence's exactness via syzygies and relates equivariant cohomology to Ext modules, extending GKM applicability.

Findings

Exactness of the sequence corresponds to a syzygy condition.

Cohomology expressed as an Ext module involving equivariant homology.

GKM method applies to Poincare duality spaces if the equivariant Poincare pairing is perfect.

Abstract

Let X be a "nice" space with an action of a torus T. We consider the Atiyah-Bredon sequence of equivariant cohomology modules arising from the filtration of X by orbit dimension. We show that a front piece of this sequence is exact if and only if the H^*(BT)-module H_T^*(X) is a certain syzygy. Moreover, we express the cohomology of that sequence as an Ext module involving a suitably defined equivariant homology of X. One consequence is that the GKM method for computing equivariant cohomology applies to a Poincare duality space if and only if the equivariant Poincare pairing is perfect.