Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2022; 59(5): 963-986

Online first article July 28, 2022      Printed September 1, 2022

https://doi.org/10.4134/JKMS.j220104

Copyright © The Korean Mathematical Society.

Weakly equivariant classification of small covers over a product of simplicies

Aslı Güçlükan İlhan, Sabri Kaan Gürbüzer

Dokuz Eylul University; Dokuz Eylul University

Abstract

Given a dimension function $\omega$, we introduce the notion of an $\omega$-vector weighted digraph and an $\omega$-equivalence between them. Then we establish a bijection between the weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of simplices $\Delta^{\omega(1)}\times\cdots \times \Delta^{\omega(m)}$ and the set of $\omega$-equivalence classes of $\omega$-vector weighted digraphs with $m$-labeled vertices, where $n$ is the sum of the dimensions of the simplicies. Using this bijection, we obtain a formula for the number of weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of three simplices.

Keywords: Small cover, weakly equivariant homeomorphism, acyclic di\-graph

MSC numbers: Primary 57S10; Secondary 37F20

Supported by: This work is supported by The Scientific and Technological Research Council of Turkey (Grant No: TBAG/118F506).

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