#### Volume 8, issue 4 (2008)

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The number of small covers over cubes

### Suyoung Choi

Algebraic & Geometric Topology 8 (2008) 2391–2399
##### 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.

##### Keywords
small cover, acyclic digraph, real torus action, equivariant homeomorphism, weak equivariant homeomorphism
##### Mathematical Subject Classification 2000
Primary: 37F20, 57S10
Secondary: 57N99
##### Publication
Received: 3 October 2008
Revised: 4 November 2008
Accepted: 13 November 2008
Published: 20 December 2008
##### Authors
 Suyoung Choi KAIST Department of Mathematical Sciences 335 Gwahangno, Yuseong-gu Daejeon, 305-701 South Korea http://topology.kaist.ac.kr/schoi