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

TL;DR

This paper develops a framework for the Duflot filtration in equivariant cohomology, applying it to group cohomology and classifying spaces, and provides new results on local cohomology modules and their properties.

Contribution

It axiomatizes the Duflot filtration for equivariant cohomology and applies it to derive new structural and local cohomology results for group cohomology.

Findings

Recovery of geometric cohomology results for classifying spaces

Construction of a chain complex for local cohomology computation

Sharper vanishing and nonvanishing results for local cohomology modules

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 $\mathbb{F}_p$ 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 $S_{p^n}$ and give vanishing and nonvanishing results for these local cohomology modules that are sharper than those given by the current theory.