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 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.

8 pages

References in corpus (1)

• Semifree circle actions, Bott towers, and quasitoric manifolds

Cited by in corpus (5)

• The number of orientable small covers over cubes
• Orientable hyperbolic 4-manifolds over the 120-cell
• Cohomological rigidity of real Bott manifolds
• An algorithmic strategy for finding characteristic maps over wedged simplicial complexes
• Crystallizations of small covers over the $n$-simplex ${\Delta }^n$ and the prism ${\Delta }^{n-1}\times I$