Brehm and K\"uhnel (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 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 traingulations of $\mathbb{HP}^2$ 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 $\mathrm{C}_3$ and $\mathrm{C}_2$, as well as the trivial symmetry group.

翻译：Brehm与Kühnel（1992）构造了三个具有对称群$\mathrm{A}_5$、$\mathrm{A}_4$和$\mathrm{S}_3$的15顶点组合8流形，其性质“类似于四元射影平面”。Gorodkov（2016）证明这三个流形实际上PL同胚于$\mathbb{HP}^2$。注意到15是与$S^8$非PL同胚的组合流形的最小顶点数。本文构造了大量$\mathbb{HP}^2$的新15顶点三角剖分。一个令人惊奇的事实是，这些例子出现在非常不同的对称群中，包括那些与群$\mathrm{A}_5$毫无关联的群。具体而言，我们发现了19个具有对称群$\mathrm{C}_7$的三角剖分、1个具有对称群$\mathrm{C}_6\times\mathrm{C}_2$的三角剖分、14个具有对称群$\mathrm{C}_6$的三角剖分、26个具有对称群$\mathrm{C}_5$的三角剖分、1个具有对称群$\mathrm{A}_4$的新三角剖分，以及11个具有对称群$\mathrm{S}_3$的新三角剖分。此外，我们得到如下分类结果：我们证明，在同构意义下，恰好存在75个具有15个顶点且对称群阶数至少为4的$\mathbb{HP}^2$三角剖分，即上述3个Brehm-Kühnel三角剖分与72个新三角剖分。另一方面，我们表明存在大量对称群为$\mathrm{C}_3$、$\mathrm{C}_2$以及平凡对称群的三角剖分。