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In 1987 Brehm and K\"uhnel showed that any combinatorial $d$-manifold withless than $3d/2+3$ vertices is PL homeomorphic to the sphere and anycombinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic toeither the sphere or a manifold like a projective plane in the sense of Eellsand Kuiper. The latter possibility may occur for $d\in\{2,4,8,16\}$ only. Thereexist a unique $6$-vertex triangulation of $\mathbb{RP}^2$, a unique $9$-vertextriangulation of $\mathbb{CP}^2$, and at least three $15$-vertex triangulationsof $\mathbb{HP}^2$. However, until now, the question of whether there exists a$27$-vertex triangulation of a manifold like the octonionic projective planehas remained open. We solve this problem by constructing a lot of examples ofsuch triangulations. Namely, we construct $634$ vertex-transitive $27$-vertexcombinatorial $16$-manifolds like the octonionic projective plane. Four of themhave symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$, andthe other $630$ have symmetry group $\mathrm{C}_3^3$ of order $27$. Further, weconstruct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial$16$-manifolds like the octonionic projective plane. Most of them have trivialsymmetry group, but there are also symmetry groups $\mathrm{C}_3$,$\mathrm{C}_3^2$, and $\mathrm{C}_{13}$. We conjecture that all thetriangulations constructed are PL homeomorphic to the octonionic projectiveplane $\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.

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