论文标题
634顶点传输，超过$ 10^{103} $非vertex-transitive-27 vertex三角形，如八元离子投影平面
634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
论文作者
论文摘要
In 1987 Brehm and Kühnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper.后者的可能性可能仅在\ {2,4,8,16 \} $中发生。存在$ \ mathbb {rp}^2 $的独特$ 6 $ vertex三角剖分，一个独特的$ 9 $ - vertex三角剖分，$ \ mathbb {cp}^2 $，以及至少三个$ 15 $ - vertex trepex triangulation of $ \ mathbb {hp} hp {hp}^2 $。但是，到目前为止，是否存在$ 27 $ vertex的三角剖分，例如八元国电射击飞机的歧管。我们通过构建许多此类三角形的例子来解决这个问题。也就是说，我们构建了$ 634 $ VERTEX传播$ 27 $ -VERTEX COMBINATORIAL $ 16 $ -MANIFOLDS，例如Octonionion射击飞机。他们中的四个具有对称组$ \ mathrm {c} _3^3 \ rtimes \ mathrm {c} _ {13} $ of $ 351 $，而其他$ 630 $具有对称组$ \ mathrm {c} _3^3 $ 27 $ 27 $。此外，我们构建了超过$ 10^{103} $非vertex-transitive $ 27 $ -VERTEX COMBINATORIAL $ 16 $ -MANIFOLDS，例如Octonionic Projective Plane。他们中的大多数都有微不足道的对称组，但是也有对称组$ \ mathrm {c} _3 $，$ \ mathrm {c} _3^2 $和$ \ mathrm {c} _ {13} $。我们猜想所有构造的三角形都是对八二世投影平面$ \ mathbb {op}^2 $的同型。然而，到目前为止，我们没有证据证明这一事实。
In 1987 Brehm and Kühnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for $d\in\{2,4,8,16\}$ only. There exist a unique $6$-vertex triangulation of $\mathbb{RP}^2$, a unique $9$-vertex triangulation of $\mathbb{CP}^2$, and at least three $15$-vertex triangulations of $\mathbb{HP}^2$. However, until now, the question of whether there exists a $27$-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct $634$ vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Four of them have symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$, and the other $630$ have symmetry group $\mathrm{C}_3^3$ of order $27$. Further, we construct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups $\mathrm{C}_3$, $\mathrm{C}_3^2$, and $\mathrm{C}_{13}$. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane $\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.
