A 15-vertex triangulation of the quaternionic projective plane

Abstract

In 1992, Brehm and Kühnel constructed a 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 $\mathbb{H}{P}^2$. 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 $M_{15}^8$. As a result, we obtain that it is indeed a minimal triangulation of $\mathbb{H}{P}^2$.

20 pages, 17 PostScript figures, text of program and raw data for the computation is included (see ancillary files)

Cited by in corpus (3)

• On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
• 634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
• New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane