A quotient criterion for syzygies in equivariant cohomology

TL;DR

This paper establishes a criterion for when the T-equivariant cohomology of a manifold with a torus action is a specific syzygy, linking it to the orbit space and stratification, thus unifying previous results.

Contribution

It provides a necessary and sufficient condition for the equivariant cohomology to be a syzygy, generalizing and unifying earlier results on freeness and torsion-freeness.

Findings

Criterion depends on the orbit space and stratification

Applicable after blowing up the non-free part of the action

Unifies previous results on equivariant cohomology

Abstract

Let X be a manifold with an action of a torus T such that all isotropy groups are connected and satisfying some other mild hypotheses. We provide a necessary and sufficient criterion for the T-equivariant cohomology of X with real coefficients to be a certain syzygy as a module over the cohomology of BT. It turns out that, possibly after blowing up the non-free part of the action, this only depends on the orbit space X/T together with its stratification by orbit type. Our criterion unifies and generalizes results of many authors about the freeness and torsion-freeness of equivariant cohomology for various classes of T-manifolds.