The number of small covers over cubes

TL;DR

This paper establishes a bijection between small covers over n-cubes and acyclic digraphs, providing formulas for counting these covers up to certain equivalences and showing bounds based on unlabeled digraphs.

Contribution

It introduces a novel bijection linking small covers over cubes to acyclic digraphs and derives counting formulas and bounds for these topological structures.

Findings

Number of small covers over n-cubes equals the number of acyclic digraphs with n labeled nodes.

Derived explicit formulas for counting small covers up to Davis-Januszkiewicz equivalence.

Proved that the count of acyclic digraphs with unlabeled nodes bounds the number of small covers up to diffeomorphism.

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 a formula of the number of small covers over an $n$-cube (generally, a product of simplices) up to Davis-Januszkiewicz equivalence classes and $\mathbf{Z}^n$-equivariant diffeomorphism 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 diffeomorphism.