A quotient criterion for syzygies in 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.

30 pages; a mistake in Proposition 3.3 of the published version is corrected, all other results remain unchanged

References in corpus (4)

• Equivariant cohomology, syzygies and orbit structure
• Syzygies in equivariant cohomology for non-abelian Lie groups
• Big polygon spaces
• Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property

Cited by in corpus (7)

• Syzygies in equivariant cohomology for non-abelian Lie groups
• Orbit spaces of equivariantly formal torus actions of complexity one
• Syzygies in equivariant cohomology in positive characteristic
• The syzygy order of big polygon spaces
• Mutants of compactified representations revisited
• The quotient criterion for syzygies in equivariant cohomology for elementary abelian $2$-group actions
• The Chang-Skjelbred lemma and generalizations