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New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane
Alexander A. Gaifullin
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Brehm and K\"uhnel (1992) constructed three 15-vertex combinatorial 8-manifolds `like the quaternionic projective plane' with symmetry groups A₅, A₄, and S₃, respectively. Gorodkov (2016) proved that these three manifolds are in fact PL homeomorphic to ℍℙ². Note that 15 is the minimal number of vertices of a combinatorial 8-manifold that is not PL homeomorphic to S⁸. In the present paper we construct a lot of new 15-vertex triangulations of ℍℙ². A surprising fact is that such examples are found for very different symmetry groups, including those not in any way related to the group A₅. Namely, we find 19 triangulations with symmetry group C₇, one triangulation with symmetry group C₆×C₂, 14 triangulations with symmetry group C₆, 26 triangulations with symmetry group C₅, one new triangulation with symmetry group A₄, and 11 new triangulations with symmetry group S₃. Further, we obtain the following classification result. We prove that, up to isomorphism, there are exactly 75 triangulations of ℍℙ² with 15 vertices and symmetry group of order at least 4: the three Brehm-K\"uhnel triangulations and the 72 new triangulations listed above. On the other hand, we show that there are plenty of triangulations with symmetry groups C₃ and C₂, as well as the trivial symmetry group.
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