On the equivariant triangulation of some small covers

TL;DR

This paper investigates equivariant triangulations of small covers, constructs minimal and unique triangulations of real projective 3-space, and develops methods for equivariant triangulations of connected sums, improving known bounds.

Contribution

It provides explicit minimal triangulations of $bR P^3$, constructs equivariant triangulations of connected sums, and establishes new lower bounds for triangulations of $bR P^4$.

Findings

Minimal $bZ_2^3$-equivariant triangulation of $bR P^3$ with 11 vertices

Unique such triangulation of $bR P^3$ with 11 vertices

Constructed $bZ_2^3$-equivariant triangulation of $bR P^3 atural bR P^3$ with 17 vertices

Abstract

In this paper, we study certain properties of $\mathbb{Z}_2^n$-equivariant triangulations of small covers. We show that any $\mathbb{Z}_2^n$-equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$, which contains $11$ vertices and prove that this is the unique $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3$ with $11$ vertices. For a finite group $G$, we give a method for constructing some $G$-equivariant triangulations of connected sums of manifolds from their respective $G$-equivariant triangulations. In particular, we construct a $\mathbb{Z}_2^3$-equivariant triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ with $17$ vertices, which is the best known yet. This triangulation of $\mathbb{RP}^3 \# \mathbb{RP}^3$ provides another minimal $g$-vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a $\ZZ_2^4$-equivariant triangulation of $\mathbb{RP}^4$ requires at least $18$ vertices.