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 $\mathbb H\mathrm P^2$. 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 $\mathbb H\mathrm P^2$. 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 $\mathbb H\mathrm P^2$ 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.