Equivariant cohomology and the Varchenko-Gelfand filtration

Daniel Moseley

Published 2011-10-24, updated 2012-05-11Version 2

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.