A note on small covers over cubes

Year 2020, , 1997 - 2006, 08.12.2020

Aslı Güçlükan İlhan

https://doi.org/10.15672/hujms.631676

Abstract

In this paper, we obtain a bijection between the weakly $\mathbb{Z}_2^n$-equivariant homeomorphism classes of small covers over an $n$-cube and the orbits of the action of $\mathbb{Z}_2 \wr S_n$ on acyclic digraphs with $n$ vertices given by local complementation and reordering of vertices. We obtain a similar formula for the number of orientable small covers over an $n$-cube. We also count the $\mathbb{Z}_2^n$-equivariant homeomorphism classes of orientable small covers and estimate the ratio between this number and the number of $\mathbb{Z}_2^n$-equivariant homeomorphism classes of small covers over an $n$-cube. 

Keywords

small cover, weakly equivariant homeomorphism, acyclic digraphs

Project Number

TBAG/118F310

References

• [1] A. Bouchet, Digraph decompositions and Eulerian systems, SIAM J. Algebraic Discrete Methods, 8, (3), 323-337, 1987.
• [2] M. Cai, X. Chen and Z. Lü, Small covers over prisms, Topol. Appl. 154, 2228–2234, 2007.
• [3] S. Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8, 2391–2399, 2008.
• [4] S. Choi, The number of orientable small covers over cubes, Proc. Japan Acad. Ser. A, 86, 97–100, 2010.
• [5] S. Choi, M. Masuda and S. Oum, Classification of real Bott manifolds and acyclic digraphs, Trans. Amer. Math. Soc. 369, 2987–3011, 2017.
• [6] M.W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus action, Duke Math. J. 62, 417–451, 1971.
• [7] D.G. Fon-Der-Flaass, Local complementations of simple and directed graphs, in: Discrete Analysis and Operations Research, 1, 15-34, 1996.
• [8] A. Garrison and R. Scott, Small covers over the dodecahedron and the 120-cell, Proc. Amer. Math. Soc. 131, 963–971, 2003.
• [9] Y. Kamishima and M. Masuda, Cohomological rigidity of real Bott manifolds, Algebr. Geom. Topol. 9, 2479-2502, 2009.
• [10] Z. Lü and M. Masuda, Equivariant classification of 2-torus manifolds, Colloq. Math. 115, 171–188, 2009.
• [11] M. Masuda, T.E. Panov, Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb. 199, 95-122. 2008.
• [12] H. Nakayama and Y. Nishimura, The orientability of small covers and coloring simple polytopes, Osaka J. Math. 42, 243–256, 2005.
• [13] V.I. Rodinov, On the number of labeled acyclic digraphs, Discrete Math. 105, 319–321, 1992.
• [14] http://users.cecs.anu.edu.au/~bdm/data/digraphs.html.