S^n中的局部凸曲线空间与群B_{n+1}的组合学
Spaces of locally convex curves in S^n and combinatorics of the group B_{n+1}
摘要：
在20世纪20年代马斯顿莫尔斯开发了什么是现在被称为莫尔斯理论试图研究拓扑空间的封闭曲线的S^2。我们提出解决一个非常相似的问题，这个问题在80年后仍然是开放的，关于S^2上的闭曲线空间的拓扑，这些闭曲线是局部凸的（即，没有拐点）。其中一个主要的困难是缺乏覆盖同伦原则的地图发送一个非封闭的局部凸曲线的Frenet框架在其端点。本文研究了S^n中具有给定初始Frenet标架和最终Frenet标架的局部凸曲线空间.利用组合数学B^{+}_{n+1} = B_{n+1} \cap SO_{n+1}，其中B_{n+1} \subset O_{n+1}是通常的Coxeter- Weyl群，我们证明了对于任何n个\ge 2，这些空间至多属于$\lceil\frac{n}{2}\rceil+1$等价类直到同胚.我们还研究了在双覆盖Spin（n+1）中的这种分类。对于$n = 2$，我们的结果完成了相应的空间分为两个拓扑不同的类，或三个类的自旋情况下的分类。
英文摘要：
In the 1920's Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S^2. We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S^2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the covering homotopy principle for the map sending a non-closed locally convex curve to the Frenet frame at its endpoint. In the present paper we study the spaces of locally convex curves in S^n with a given initial and final Frenet frames. Using combinatorics of B^{+}_{n+1} = B_{n+1} \cap SO_{n+1}, where B_{n+1} \subset O_{n+1} is the usual Coxeter-Weyl group, we show that for any n \ge 2 these spaces fall in at most $\lceil\frac{n}{2}\rceil+1$ equivalence classes up to homeomorphism. We also study this classification in the double cover Spin(n+1). For $n = 2$ our results complete the classification of the corresponding spaces into two topologically distinct classes, or three classes in the spin case.
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