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 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. (c) 2006 Elsevier Inc. All rights reserved.