Syzygies in equivariant cohomology for non-abelian Lie groups

Abstract

We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.

28 pages; minor changes

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• Equivariant cohomology, syzygies and orbit structure
• A quotient criterion for syzygies in equivariant cohomology
• Big polygon spaces
• Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property
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Cited by in corpus (7)

• A quotient criterion for syzygies in equivariant cohomology
• Syzygies in equivariant cohomology in positive characteristic
• The equivariant cohomology for semidirect product actions
• The quotient criterion for syzygies in equivariant cohomology for elementary abelian $2$-group actions
• On Equivariant Poincaré Duality, Gysin Morphisms and Euler Classes
• Equivariant cohomology of cohomogeneity one actions: the topological case
• The Chang-Skjelbred lemma and generalizations