Cleanliness and the Varchenko-Gelfand algebra

Abstract

A central question in the theory of hyperplane arrangements is when the complement of a complex arrangement is aspherical. Barkley and Speyer introduced a class of real arrangements that are called "clean," and Yoshinaga proved that every real arrangement whose complexification is $K(\pi,1)$ is clean. We show that cleanliness is equivalent to a natural statement about the Varchenko-Gelfand ring, which in practice allows for fast calculation. We conclude with an investigation of the relationships between various properties of arrangements, including cleanliness and the asphericity of the arrangement complement.