The number of small covers over cubes

Abstract

In the present paper we find a bijection between the set of small covers over an n-cube and the set of acyclic digraphs with n labeled nodes. Using this, we give formulas of the number of small covers over an n-cube (generally, a product of simplices) up to Davis-Januszkiewicz equivalence classes and Z(2)(n)-equivariant homeomorphism classes. Moreover we prove that the number of acyclic digraphs with n unlabeled nodes is an upper bound of the number of small covers over an n-cube up to homeomorphism.