A new family of triangulations of $\mathbb{R}P^d$

TL;DR

This paper introduces a new family of piecewise-linear triangulations for the real projective space $\\mathbb{R}P^d$, significantly reducing the number of vertices needed compared to previous constructions.

Contribution

It presents a novel construction of PL triangulations of $\\mathbb{R}P^d$ with fewer vertices, improving upon K"{u}hnel's earlier method.

Findings

Triangulations on the order of \\left(\frac{1+\sqrt{5}}{2}\right)^{d+1} vertices

Reduction from previous $2^{d+1}-1$ vertices

Applicable for all dimensions $d \geq 1$

Abstract

We construct a family of PL triangulations of the $d$-dimensional real projective space $\mathbb{R}P^d$ on $\Theta((\frac{1+\sqrt{5}}{2})^{d+1})$ vertices for every $d\geq 1$. This improves a construction due to K\"{u}hnel on $2^{d+1}-1$ vertices.