The equivariant Orlik-Solomon algebra

الأصل · EN

Given a real arrangement A, the complement M(A) of the complexification of A admits an action of 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 Z₂. We give a combinatorial presentation of this ring, and interpret it as a deformation of the ordinary Orlik-Solomon algebra into the Varchenko-Gel'fand ring of locally constant Z₂-valued functions on the complement C(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.