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

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Abstract: In 1987 Brehm and K"uhnel showed that any combinatorial

d

-manifold with less than

3 d / 2 + 3

vertices is PL homeomorphic to the sphere and any combinatorial

d

-manifold with exactly

3 d / 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 i n 2, 4, 8, 16

only. There exist a unique

6

-vertex triangulation of

m a t h b b {R P}^{2}

, a unique

9

-vertex triangulation of

m a t h b b {C P}^{2}

, and at least three

15

-vertex triangulations of

m a t h b b {H P}^{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

m a t h r m C_{3}^{3} t i m e s m a t h r m C_{13}

of order

351

, and the other

630

have symmetry group

m a t h r m C_{3}^{3}

of order

27

. Further, we construct more than

1 0^{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

m a t h r m C_{3}

,

m a t h r m C_{3}^{2}

, and

m a t h r m C_{13}

. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane

m a t h b b {O P}^{2}

. Nevertheless, we have no proof of this fact so far.

Has companion code repository: https://github.com/agaif/triangulations-like-op2

Mathematics Subject Classification ID

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