On manifolds defined by 4-colourings of simple 3-polytopes
On manifolds defined by 4-colourings of simple 3-polytopes
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关于由简单 3 多胞形的 4 着色定义的流形
DOI：
10.1070/rm9738
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发表时间：
2016
影响因子：
0.9
通讯作者：
T. Panov
中科院分区：
文献类型：
--
作者：
V. Buchstaber;T. Panov
令 $\mathcal{P}$ 为组合 3 维简单多面体 $P$ 的类，与四面体不同，没有 3 和 4 面带。根据 Pogorelov 和 Andreev 的结果，多面体 $P$ 承认在 Lobachevsky 空间 $\mathbb{L}^3$ 中具有直二面角的实现当且仅当 $P \in \mathcal{P}$ 时。我们考虑由 Pogorelov 多胞体 P 的正则 4 着色定义的两个光滑流形族：$P$ 上的六维准流形和 $P$ 的三维小覆盖；后者也称为 Loebell 型三维双曲流形。我们证明来自任一族的两个流形是微分同胚的当且仅当相应的 4 色是等价的。
Let $\mathcal{P}$ be the class of combinatorial 3-dimensional simple polytopes $P$, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope $P$ admits a realisation in Lobachevsky space $\mathbb{L}^3$ with right dihedral angles if and only if $P \in \mathcal{P}$. We consider two families of smooth manifolds defined by regular 4-colourings of Pogorelov polytopes P: six-dimensional quasitoric manifolds over $P$ and three-dimensional small covers of $P$; the latter are also known as three-dimensional hyperbolic manifolds of Loebell type. We prove that two manifolds from either of the families are diffeomorphic if and only if the corresponding 4-colourings are equivalent.