A note on small covers over cubes

Year 2020, Volume: 49 Issue: 6, 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

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