An Efficient Triangulation of $\mathbb{R}P^5$

TL;DR

This paper introduces a highly symmetric 6-dimensional polytope that provides a minimal vertex triangulation of $\,\mathbb{R}P^5$, and also improves triangulation bounds for $\,\mathbb{R}P^6$.

Contribution

It presents a new minimal vertex triangulation of $\,\mathbb{R}P^5$ and improved triangulations of $\,\mathbb{R}P^6$, advancing the understanding of projective space triangulations.

Findings

Constructed a 6-polytope with 24 vertices for $\,\mathbb{R}P^5$

Produced $\,\mathbb{R}P^6$ triangulations with 45 and 49 vertices

Improved previous vertex bounds for $\,\mathbb{R}P^6$ triangulations

Abstract

We present a $6$-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a $24$-vertex triangulation of the $5$-dimensional real projective space. This $6$-polytope is highly symmetric with an automorphism group of order $192$, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of $\mathbb{R}P^5$. Our method also produces two triangulations of $\mathbb{R}P^6$ on $45$ and $49$ vertices; both improve the previously best known construction in dimension $6$ that used $53$ vertices.