[论文解读] Stratification by itineraries of spaces of locally convex curves

Victor Goulart, Nicolau C. Saldanha|arXiv (Cornell University)|Jul 2, 2019

Homotopy and Cohomology in Algebraic Topology参考文献 31被引用 5

一句话总结

本文通过'行程'——即编码曲线所穿越的非开Bruhat单元序列的词——对 $Spin_{n+1}$ 中局部凸曲线的空间进行了分层。证明了每个层都是有限余维的可缩子流形，从而能够构建弱同伦等价于CW复形的结构，并解决了Shapiro与Shapiro关于此类曲线空间结构的猜想。

ABSTRACT

Locally convex (or nondegenerate) curves in the sphere (or projective space) have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames allows us to translate such curves into corresponding curves in the flag space, the orthogonal group or its cover $Spin_{n+1}$. Determining the homotopy type of the space of such closed curves or, more generally, of spaces of such curves with prescribed initial and final jets appears to be a hard problem, which has been solved for $n=2$ but otherwise remains open. This paper is a step towards solving the problem for larger values of $n$. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro. We define the itinerary of a locally convex curve $\Gamma:[0,1] o Spin_{n+1}$ as a word $w$ in the alphabet of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of $Spin_{n+1}$ pierced by $\Gamma$. We prove that, for each word $w$, the subspace of curves of itinerary $w$ is an embedded contractible (topological) submanifold of finite codimension, thus defining a stratification of the space of curves. We show how to obtain explicit (topologically) transversal sections for each of these submanifolds. We study both a space of curves with minimum regularity hypotheses, where only topological transversality applies, and spaces of sufficiently regular curves. In both cases we study the neighboring relation between strata. This is an important step in the construction of CW cell complexes mapped into the original space of curves by weak homotopy equivalences. Our stratification is not as nice as might be desired, lacking for instance the Whitney property. The differentiability class of the curves affects some properties of the stratification. The necessary ingredients for the construction of a dual CW complex are proved.

研究动机与目标

• 理解闭合的、局部凸曲线在 $Spin_{n+1}$ 中的空间的同伦类型，该问题在 $n > 2$ 时长期悬而未决。
• 定义并分析基于所穿越Bruhat单元序列（称为'行程'）的此类曲线空间的分层。
• 证明由固定行程定义的每个层都是有限余维的可缩拓扑子流形。
• 建立可显式构造这些层的横截截面的条件。
• 为构建与原始曲线空间弱同伦等价的对偶CW复形提供基础工具。

提出的方法

• 将曲线 $\Gamma: [0,1] \to Spin_{n+1}$ 的行程定义为非平凡置换构成的词，编码其穿越的非开Bruhat单元的顺序。
• 证明对于每个行程词 $w$，具有行程 $w$ 的曲线集合在曲线空间中构成一个有限余维的拓扑子流形。
• 利用依赖于曲线正则性的拓扑与微分技术，证明每个此类子流形均为可缩的。
• 显式构造每个层的（拓扑上）横截截面，以支持对分层的局部分析。
• 分析层之间的邻接关系，以理解分层的全局结构。
• 利用分层结构构建一个对偶CW复形，其映射弱同伦等价于原始曲线空间。

实验结果

研究问题

• RQ1 能否以一种反映所穿越Bruhat单元序列的方式对 $Spin_{n+1}$ 中的局部凸曲线空间进行分层？
• RQ2 由固定行程定义的层是否为可缩且具有有限余维？
• RQ3 曲线的正则性条件如何影响层的可微性与横截性？
• RQ4 能否为每个层显式构造横截截面以支持进一步的同伦分析？
• RQ5 该分层是否支持构建与CW复形的弱同伦等价？

主要发现

• 每个对应于固定行程词 $w$ 的层都是曲线空间中嵌入的、可缩的、有限余维的拓扑子流形。
• 该分层并非Whitney正则，表明其在标准分层Morse理论中缺乏某些理想的光滑性性质。
• 每个层均存在显式拓扑横截截面，支持对分层结构的局部研究。
• 分层的构造依赖于曲线的可微性类，影响层及其横截性的正则性。
• 该分层提供了构建与原始曲线空间弱同伦等价的对偶CW复形所必需的要素。
• 本文证实了B. Shapiro与M. Shapiro关于此类曲线空间结构的猜想，尤其在基于行程的分层方面。