Mathematical Sciences: Topology of Foliations and Foliated Knot Complements

数学科学：叶状拓扑和叶结补集

• 批准号: 9201723
• 负责人: Lawrence Conlon
• 金额: $ 12.12万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-15 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201723&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Foliations; Foliated

This project concerns a long-standing collaboration with John Cantwell into the structure of foliations of codimension-1, especially on closed 3-manifolds. It grows directly out of their theory of levels (1978), which has influenced not only their own work, but that of the French and Japanese schools as well. The main direction will be to explore the deep relations between the topology of knot complements and taut, finite depth foliations. A major tool in this investigation will be the "generalized Poincare- Bendixson" theory developed by the investigator and Cantwell as part of the theory of levels. The lowest level of complexity is that of a fibered knot, here called a "depth-zero knot." These knots are relatively rare, have been studied classically, and are relatively well understood. The next level of complexity is exhibited by the depth-one knots, which are very numerous (as shown in David Gabai's thesis) and present many interesting questions. Depth-one knots are currently the focus of this project, and results already obtained confirm their interest. Other questions to be studied relate the degree of smoothability of taut foliations to topology. Although smoothness questions have not particularly concerned 3-manifold topologists, a strong case can be made that their relevance to topology is deep. In a rather different direction, the proposer remains interested in the ergodic theory of foliations, especially of exceptional minimal sets, and the related question of whether the Godbillon-Vey invariant detects such minimal sets (almost certainly not). A foliation of a manifold is a way of filling the manifold with lower dimensional pieces. In the case of a codimension-one foliation, these pieces are of dimension one less than that of the given manifold. Think of an onion or an artichoke. The topology of a manifold is strongly related to the kind of foliation which it will support, and in skillful hands this relation has been forged into a powerful tool for investigating the topology of manifolds. It is an unintuitive fact that the major algebraic tools for investigating the topology of manifolds work best in the case of high dimensional manifolds. The geometric tool afforded by foliations is thus particularly welcome in the case of low dimensional manifolds. A major instance of this is the investigation of the complement of a knot in the three-dimensional sphere, which turns out to be an important way to gain information about the knot itself.

该项目涉及与约翰的长期合作 Cantwell引入余维1的叶理结构， 尤其是在闭三维流形上。 它直接生长在 层次理论（1978年），这不仅影响了他们自己的 这是法国和日本的学校，也是。 的 主要方向将是探索之间的深层关系 拓扑结补集和紧张，有限深度叶理。 一 这项研究的主要工具将是“广义庞加莱- Bendixson”理论是由调查员和Cantwell提出的， 层次理论的一部分。 最低的复杂程度是 即纤维结，这里称为“零深度结"。“这些 结是相对罕见的，已经被经典地研究过， 相对比较好理解。 下一个层次的复杂性是 由深度一节展示，这是非常多的（如图所示 在大卫Gabai论文中），并提出了许多有趣的问题。 深度一节是目前该项目的重点， 已经取得的成果证实了他们的兴趣。 其他问题 与绷紧叶理的光滑程度有关 到拓扑学 虽然平滑度问题还没有特别 关于3-流形拓扑学家，一个强有力的情况下， 它们与拓扑学的相关性很深。 在一个相当不同的 方向，提议者仍然对遍历理论感兴趣， 叶理，特别是例外极小集，以及相关的 问题是Godbillon-Vey不变量是否检测到这种 最小集（几乎肯定不是）。 流形的叶理是填充流形的一种方法 低维度的碎片。 在余维1的情况下， 叶理，这些碎片的尺寸小于 给定流形。 想想洋葱或朝鲜蓟。 拓扑 一个流形的形状与它所处的叶理类型密切相关， 将支持，并在熟练的手，这种关系已经锻造 成为研究流形拓扑的有力工具。 这是一个不直观的事实，主要的代数工具， 研究流形的拓扑结构在以下情况下效果最好： 高维流形 提供的几何工具， 因此，在低海拔地区， 维流形 这方面的一个主要例子是 三维空间中纽结的补集研究 球体，这被证明是获取信息的重要方式 关于绳结本身