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 \(\mathbb H\mathrm P^2\) . 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.
