The equivariant Orlik–Solomon algebra

Given a real hyperplane arrangement A, the complement M(A) of the complexification of A admits an action of the group Z2 by complex conjugation. We define the equivariant Orlik–Solomon algebra of A to be the Z2-equivariant cohomology ring of M(A) with coefficients in the field F2. 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 F2-valued functions on the complement MR(A) of A in Rn. We also show that the Z2-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.