The equivariant Orlik-Solomon algebra

Given a real arrangement $A$, the complement $M(A)$ of the complexification of $A$ admits an action of ${\mathbb{Z}}_2$ by complex conjugation. We define the equivariant Orlik-Solomon algebra of $A$ to be the ${\mathbb{Z}}_2$-equivariant cohomology ring of $M(A)$ with coefficients in ${\mathbb{Z}}_2$. 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 $\text{mattbb}{Z}_2$-valued functions on the complement $C(A)$ of $A$ in ${\mathbb{R}}^n$. We also show that the ${\mathbb{Z}}_2$-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 $A^{\mathrm{\prime }}$ such that $M(A)$ and $M(A^{\mathrm{\prime }})$ have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik-Solomon algebra.