Cleanliness and the Varchenko-Gelfand algebra

TL;DR

This paper explores the concept of cleanliness in hyperplane arrangements, establishing its equivalence to a property of the Varchenko-Gelfand ring, and investigates its relation to asphericity of arrangement complements.

Contribution

It demonstrates that cleanliness is equivalent to a natural property of the Varchenko-Gelfand ring, enabling efficient calculations and linking it to the asphericity of arrangement complements.

Findings

Clean arrangements are characterized by a property of the Varchenko-Gelfand ring.

Cleanliness is equivalent to a natural statement about the Varchenko-Gelfand algebra.

Clean arrangements relate to the asphericity of the arrangement complement.

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.