The equivariant Orlik–Solomon algebra

Abstract

Given a real hyperplane arrangement A , the complement M ( A ) of the complexification of A admits an action of the group Z 2 by complex conjugation. We define the equivariant Orlik–Solomon algebra of A to be the Z 2 -equivariant cohomology ring of M ( A ) with coefficients in the field F 2 . We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik–Solomon algebra into the Varchenko–Gelfand ring of locally constant F 2 -valued functions on the complement M R ( A ) of A in R n . We also show that the Z 2 -equivariant homotopy type of M ( A ) is determined by the oriented matroid of A . As an application, we give two examples of pairs of arrangements A and A ′ such that M ( A ) and M ( A ′ ) have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik–Solomon algebra.