Equivariant cohomology, syzygies and orbit structure

Details Equivariant cohomology, syzygies and orbit structure Equivariant cohomology, syzygies and orbit structure

2011-11-03

arxiv.org/abs/1111.0957v1

Mathematics

Algebraic Topology

28 pages

Scientific paper

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. This can be combined with an equivariant version of Poincare-Alexander-Lefschetz duality, which we also establish. 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.

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