On a vertex-minimal triangulation of $\mathbb{R}P^4$

TL;DR

This paper presents three constructions of a vertex-minimal triangulation of 4-dimensional real projective space, improving understanding of minimal triangulations and related combinatorial structures.

Contribution

It introduces three new constructions of a minimal triangulation of  real projective space, including a symmetric 32-vertex sphere and approaches to reduce vertex count.

Findings

Constructed a 32-vertex triangulation with high symmetry

Provided methods to improve vertex bounds for  space triangulations

Connected the triangulation to the 22-point Witt design

Abstract

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}P^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated $\mathbb{R}P^4$ and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate $n$-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known $16$-vertex triangulation of $\mathbb{R}P^4$. We also give a short, simple construction of the $22$-point Witt design, which is closely related to the complex we construct.