A 15-Vertex Triangulation of the Quaternionic Projective Plane

Abstract

In 1992, Brehm and Kühnel 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 HP 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 HP 2 .

Generalized Heawood Numbers

“…2. Three distinct triangulations of the quaternionic projective plane HP 2 with 15 vertices [12,24], among them one with a vertex transitive group action of A 5 .…”

Combinatorics of unavoidable complexes

“…From the list of 11 exceptional, vertex-minimal triangulations of manifolds exhibited in [L, Table 2] we select the 'projective planes' {[3], RP 2 6 , CP 2 9 , HP 2 15 }, since they are all Alexander self-dual complexes (π(K) = 2) and have a vertex-transitive group of symmetry (which makes the ρ-invariant easily computable). More information about these important complexes can be found in [KB83,BD94,BK92,Go].…”

Estimates of covering type and minimal triangulations based on category weight

“…Even when there is an explicit triangulation at hand it may be difficult to show that it represents a specific manifold. This was the case of the Brehm-Kühnel triangulation [5] whose cohomology is that of the quaternionic projective plane, but it required a very hard computation with combinatorial Pontrjagin classes by Gorodkov [13] to show that it is indeed the minimal triangulation of HP 2 . This explains why apart from classical minimal triangulations of spheres and closed surfaces, and a special family of minimal triangulations for certain sphere bundles over a circle (so called Császár tori -see Kühnel [22]), there exists only a handful of examples for which the minimal triangulations are known.…”