Algebraic & Geometric Topology

The number of small covers over cubes

Suyoung Choi

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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 2n–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.

Article information

Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2391-2399.

Received: 3 October 2008
Revised: 4 November 2008
Accepted: 13 November 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F20: Combinatorics and topology 57S10: Compact groups of homeomorphisms
Secondary: 57N99: None of the above, but in this section

small cover acyclic digraph real torus action equivariant homeomorphism weak equivariant homeomorphism


Choi, Suyoung. The number of small covers over cubes. Algebr. Geom. Topol. 8 (2008), no. 4, 2391--2399. doi:10.2140/agt.2008.8.2391.

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