A note on small covers over cubes

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Hacettepe Journal of Mathematics and Statistics
Volume:49 Issue:6
A note on small covers over cubes

A note on small covers over cubes

Pages : 1997-2006

View : 125 | Download : 9

Publication Date : 2020-12-08

Article Type : Research Paper

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

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