On the Duflot filtration for equivariant cohomology rings and applications to group cohomology

2017/11/15 by James C. Cameron, Cameron, James C.
Mathematics · #55N91 (Primary) 20J06 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1711.05832

openalex publication_date 2017/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Duflot filtration on the Borel equivariant cohomology of smooth manifolds with a smooth p-torus action. We axiomatize the filtration and prove analog of several structural results about equivariant cohomology rings in this setting. We apply this abstract theory to study the \mathbbF_p cohomology rings of classifying spaces of compact Lie groups, and show how to recover geometric results about the cohomology of BG using equivariant cohomology. This includes some results about detection on subgroups and restrictions on associated primes that were previously only known for finite groups. We are particularly interested in the local cohomology modules of equivariant cohomology rings, and we construct a tractable chain complex computing local cohomology. As an application, we study the local cohomology of the group cohomology of the p-Sylow subgroups of S_pⁿ and give vanishing and nonvanishing results for these local cohomology modules that are sharper than those given by the current theory.

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