Equivariant cohomology, syzygies and orbit structure

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.

23 pages. The former Section 6 has been substantially expanded and is now a separate paper, see arXiv:1303.1146. Several minor changes

Cited by in corpus (9)

• Syzygies in equivariant cohomology for non-abelian Lie groups
• A quotient criterion for syzygies in equivariant cohomology
• Big polygon spaces
• Syzygies in equivariant cohomology in positive characteristic
• Symmetric products of equivariantly formal spaces
• Self-dual binary codes from small covers and simple polytopes
• Mutants of compactified representations revisited
• The syzygy order of big polygon spaces
• The Chang-Skjelbred lemma and generalizations