634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

Abstract

In 1987 Brehm and Kühnel showed that any combinatorial d-manifold with less than 3d/2 + 3 vertices is PL homeomorphic to the sphere and any combinatorial d-manifold with exactly 3d/2 + 3 vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for d ∈ {2, 4, 8, 16} only. There exist a unique 6-vertex triangulation of RP 2 , a unique 9-vertex triangulation of CP 2 , and at least three 15-vertex triangulations of HP 2 . However, until now, the question of whether there exists a 27-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct 634 vertex-transitive 27-vertex combinatorial 16-manifolds like the octonionic projective plane. Four of them have symmetry group C 3 3 ⋊ C 13 of order 351, and the other 630 have symmetry group C 3 3 of order 27. Further, we construct more than 10 103 non-vertex-transitive 27-vertex combinatorial 16-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups C 3 , C 2 3 , and C 13 . We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane OP 2 . Nevertheless, we have no proof of this fact so far.

Generalized Heawood Numbers

“…, f k and χ(M ). If one puts in 5. k = 8: Many 27-vertex triangulations of a 16-manifold "like the octonionic plane", see [22].…”

“…In particular the 2-skeleton of the 8-simplex embeds into the complex projective plane and the 4-skeleton of the 14-simplex embeds into the quaternionic projective plane. Moreover the 8-skeleton of the 26-simplex embeds into certain manifolds "like the Cayley plane" [22].…”