New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane

Abstract

Brehm and Kühnel (1992) constructed three 15-vertex combinatorial 8-manifolds `like the quaternionic projective plane' with symmetry groups ${\mathrm{A}}_5$, ${\mathrm{A}}_4$, and ${\mathrm{S}}_3$, respectively. Gorodkov (2016) proved that these three manifolds are in fact PL homeomorphic to ${\mathbb{HP}}^2$. Note that 15 is the minimal number of vertices of a combinatorial 8-manifold that is not PL homeomorphic to $S^8$. In the present paper we construct a lot of new 15-vertex triangulations of ${\mathbb{HP}}^2$. A surprising fact is that such examples are found for very different symmetry groups, including those not in any way related to the group ${\mathrm{A}}_5$. Namely, we find 19 triangulations with symmetry group ${\mathrm{C}}_7$, one triangulation with symmetry group ${\mathrm{C}}_6\times {\mathrm{C}}_2$, 14 triangulations with symmetry group ${\mathrm{C}}_6$, 26 triangulations with symmetry group ${\mathrm{C}}_5$, one new triangulation with symmetry group ${\mathrm{A}}_4$, and 11 new triangulations with symmetry group ${\mathrm{S}}_3$. Further, we obtain the following classification result. We prove that, up to isomorphism, there are exactly 75 triangulations of ${\mathbb{HP}}^2$ with 15 vertices and symmetry group of order at least 4: the three Brehm-Kühnel triangulations and the 72 new triangulations listed above. On the other hand, we show that there are plenty of triangulations with symmetry groups ${\mathrm{C}}_3$ and ${\mathrm{C}}_2$, as well as the trivial symmetry group.

41 pages, minor changes

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