Yoshinaga's criterion 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. Yoshinaga provides a necessary condition in the case of complexifications of real arrangements, which he formulates purely combinatorially. We show that Yoshinaga's criterion 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 Yoshinaga's criterion and the asphericity of the arrangement complement.