Equivariant cohomology and the Varchenko-Gelfand filtration

TL;DR

This paper extends the understanding of equivariant cohomology for configuration spaces and hyperplane arrangements, providing new algebraic presentations and generalizations to oriented matroids.

Contribution

It generalizes previous results on symmetric group actions and cohomology of configuration spaces to affine arrangements and oriented matroids, with explicit cohomology ring presentations.

Findings

Cohomology of configuration spaces is isomorphic to the regular representation.

Extended equivariant cohomology results to affine subspace arrangements.

Provided presentations of the equivariant cohomology ring.

Abstract

The cohomology of the configuration space of n points in R^3 admits a symmetric group action and has been shown to be isomorphic to the regular representation. One way to prove this is by defining an S^1-action whose fixed point set is the complement of the braid arrangement and using equivariant cohomology to show that the cohomology of the configuration space is the associated graded algebra of the cohomology of the arrangement complement. In this paper, we will extend this result to the setting of affine subspace arrangements coming from real hyperplane arrangements using similar methods. We also provide a presentation of the equivariant cohomology ring and extend some results to the setting of oriented matroids.