A 15-Vertex Triangulation of the Quaternionic Projective Plane

In 1992, Brehm and Kühnel constructed an 8-dimensional simplicial complex M 15 8 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 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 15 8 . As a result, we obtain that it is indeed a minimal triangulation of H P 2 .