Equilibrium and equivariant triangulations of some small covers with minimum number of vertices

TL;DR

This paper introduces equilibrium triangulations for small covers, focusing on minimal vertex triangulations of 2D small covers and specific 3D manifolds, using the theory of small covers.

Contribution

It defines equilibrium triangulations for small covers and constructs minimal vertex triangulations for certain 3-manifolds and real projective spaces.

Findings

Vertex minimal equilibrium triangulations of specific 3-manifolds constructed.

Equilibrium triangulations of $ ext{RP}^n$ with $2^n + n + 1$ vertices provided.

Analysis of $ ext{Z}_2^2$-equivariant triangulations of 2D small covers.

Abstract

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of $2$-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{RP}^3 \# \mathbb{RP}^3$, $S^1 \times \mathbb{RP}^2$ and a nontrivial $S^1$ bundle over $\mathbb{RP}^2$. We construct some nice equilibrium triangulations of the real projective space $\mathbb{RP}^n$ with $2^n +n+1$ vertices. The main tool is the theory of small covers.