Mathematical Sciences: Topology of Foliations and Foliated Knot Complements

这个项目涉及与John Cantwell在余维-1的叶状结构方面的长期合作，特别是在封闭的3-流形上。它直接源于他们的层次理论(1978)，该理论不仅影响了他们自己的工作，也影响了法国和日本学派的工作。主要方向将是探索纽结补的拓扑和紧的、有限深度的叶层之间的深层次关系。这项研究的一个主要工具将是研究人员和坎特维尔作为水平理论的一部分发展起来的“广义Poincare-Bendixson”理论。复杂程度最低的是纤维结，这里称为“深度零结”。这些结相对罕见，已经被经典地研究过，并且相对地被很好地理解。下一层次的复杂性表现为深度一节，它非常多(如大卫·加拜的论文所示)，并提出了许多有趣的问题。深度一节是目前这个项目的重点，已经获得的结果证实了他们的兴趣。其他要研究的问题是紧致叶的光滑度与拓扑学有关。虽然光滑性问题并没有引起三维流形拓扑学家的特别关注，但有一个强有力的例子表明，它们与拓扑学的相关性是很深的。在一个相当不同的方向上，提出者仍然对叶的遍历理论感兴趣，特别是例外极小集的遍历理论，以及相关的问题，即戈德韦恩不变量是否能检测到这样的极小集(几乎肯定不能)。流形的叶状结构是用较低维度的部件填充流形的一种方式。在余维一叶理的情况下，这些片断的尺寸比给定流形的尺寸小一。想想洋葱或洋蓟。流形的拓扑与它所支持的叶层的种类密切相关，在熟练的人手中，这种关系已经被锻造成研究流形拓扑的有力工具。一个不直观的事实是，研究流形拓扑的主要代数工具在高维流形的情况下工作得最好。因此，在低维流形的情况下，由叶提供的几何工具特别受欢迎。这方面的一个主要例子是研究三维球面中的纽结的补集，这被证明是获得关于纽结本身的信息的重要途径。

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.