Equivariant cohomology and the Varchenko–Gelfand filtration

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The cohomology of the configuration space of n points in R3 is isomorphic to the regular representation of the symmetric group, which acts by permuting the points. We give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the Varchenko–Gelfand filtration on the cohomology of the configuration space of n points in R1. Along the way, we give a presentation of the equivariant cohomology ring of the R3 configuration space with respect to a circle acting on R3 via rotation around a fixed line. We extend our results to the settings of arbitrary real hyperplane arrangements (the aforementioned theorems correspond to the braid arrangement) as well as oriented matroids.