The equivariant Orlik-Solomon algebra

TL;DR

This paper introduces the equivariant Orlik-Solomon algebra for real arrangements, providing a combinatorial presentation and showing its ability to distinguish arrangements with identical non-equivariant homotopy types.

Contribution

It defines a new equivariant cohomology ring for arrangements and interprets it as a deformation of the classical Orlik-Solomon algebra, linking it to the Varchenko-Gel'fand ring.

Findings

Provides a combinatorial presentation of the equivariant Orlik-Solomon algebra.

Shows the algebra as a deformation of the classical Orlik-Solomon algebra.

Demonstrates the algebra's ability to distinguish arrangements with identical non-equivariant homotopy types.

Abstract

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 $\mattbb{Z}_2$-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 $A'$ such that $M(A)$ and $M(A')$ have the same nonequivariant homotopy type, but are distinguished by the equivariant Orlik-Solomon algebra.