A 15-Vertex Triangulation of the Quaternionic Projective Plane

Abstract

In 1992, Brehm and Kühnel constructed an 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 \({\mathbb {H}}P^2\).