A 15-vertex triangulation of the quaternionic projective plane

TL;DR

This paper confirms that a 15-vertex simplicial complex is a minimal triangulation of the quaternionic projective plane by computing its first Pontryagin class, resolving a long-standing question in topological combinatorics.

Contribution

The authors successfully compute the first Pontryagin class of a 15-vertex complex, proving it is PL homeomorphic to the quaternionic projective plane, establishing it as a minimal triangulation.

Findings

Confirmed the complex is a minimal triangulation of P^2

Computed the first Pontryagin class using Gaifullin's algorithm

Resolved the topological classification of the complex

Abstract

In 1992, Brehm and K\"uhnel constructed a 8-dimensional simplicial complex $M^8_{15}$ with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold "like a projective plane" in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to $\mathbb{H}P^2$. This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of $M^8_{15}$. As a result, we obtain that it is indeed a minimal triangulation of $\mathbb{H}P^2$.