Weakly Equivariant Classification of Small Covers over a Product of Simplicies

TL;DR

This paper introduces a new combinatorial approach using weighted digraphs to classify small covers over products of simplices, establishing a bijection with certain equivalence classes.

Contribution

It defines $ ext{ extomega}$-vector weighted digraphs and proves a bijection with weakly equivariant homeomorphism classes of small covers over product simplices.

Findings

Established a bijection between homeomorphism classes and $ extomega$-equivalence classes.

Derived a formula for counting classes over a product of three simplices.

Provided a combinatorial framework for classifying small covers.

Abstract

Given a dimension function $\omega$, we define a 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 $\Delta^{n_1}\times\cdots \times \Delta^{n_k}$ and the set of $\omega$-equivalence classes of $\omega$-vector weighted digraphs with $k$-labeled vertices. As an example, 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.