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

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.