论文标题
简单孔的拓扑和旗帜的拓扑和几何形状
Topology and geometry of flagness and beltness of simple orbifolds
论文作者
论文摘要
我们考虑一类右角的Coxeter Orbifolds，称为简单的Orbifolds，它们是简单多面体的概括。与简单的多面体上的歧管类似，歧管上的歧管的拓扑和几何形状与简单孔的组合物和Orbifold结构密切相关。我们将简单多型的旗帜和皮带的概念概括为简单的Orbifolds的设置。 为了描述一个简单的Orbifold的拓扑结构和几何形状，我们专注于{\ em简单的把手}（也就是说，可以通过粘合一些脱节特定的Codimension-One Faces来从简单的多型中获得的简单圆锥形）。我们在组合学方面证明了以下两个主要结果，可以将其理解为简单手柄上的“组合球理定理”和“组合平面圆环定理”： （a）一个简单的句柄是在且仅当它是标志的情况下是偏差的。 （b）在$π_1^{orb}（q）（q）的$π_1^{orb}（q）$时，存在一个cart-two，仅当它包含$ \ square $ - belt时，就有一个olbifold-spherical简单的句柄$ q $。 此外，根据这两个结果和一些几何结果，可以表明，在简单的句柄$ q $上的一定歧管盖上某些弯曲（歧管double）上的存在可以根据$ q $的组合来表征。在三维情况下，连同双曲线3个序曲理论，我们可以为简单的$ 3 $ handleboby诱导纯粹的组合等效描述，以承认右角屈曲结构，这是Pogorelov Theorem的自然概括。
We consider a class of right-angled Coxeter orbifolds, named as simple orbifolds, which are a generalization of simple polytopes. Similarly to manifolds over simple polytopes, the topology and geometry of manifolds over simple orbifolds are closely related to the combinatorics and orbifold structure of simple orbifolds. We generalize the notions of flag and belt in the setting of simple polytopes into the setting of simple orbifolds. To describe the topology and geometry of a simple orbifold in terms of its combinatorics, we focus on {\em simple handlebodies} (that is, simple orbifolds which can be obtained from simple polytopes by gluing some disjoint specific codimension-one faces). We prove the following two main results in terms of combinatorics, which can be understood as "Combinatorial Sphere Theorem" and "Combinatorial Flat Torus Theorem" on simple handlebodies: (A) A simple handlebody is orbifold-aspherical if and only if it is flag. (B) There exists a rank-two free abelian subgroup in $π_1^{orb}(Q)$ of an orbifold-aspherical simple handlebody $Q$ if and only if it contains an $\square$-belt. Furthermore, based on such two results and some results of geometry, it is shown that the existence of some curvatures on a certain manifold cover (manifold double) over a simple handlebody $Q$ can be characterized in terms of the combinatorics of $Q$. In 3-dimensional case, together with the theory of hyperbolic 3-manifolds, we can induce a pure combinatorial equivalent description for a simple $3$-handlebody to admit a right-angled hyperbolic structure, which is a natural generalization of Pogorelov Theorem.
