Mathematical Sciences: Topology of Foliations and Foliated Knot Complements

该项目涉及与John Cantwell长期合作的余维-1叶的结构，特别是在封闭的3流形上。它直接源于他们的层次理论（1978），这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。主要的方向将是探索结补的拓扑结构与紧致的有限深度叶理之间的深层关系。这项研究的一个主要工具将是由研究者和坎特韦尔作为水平理论的一部分发展起来的“广义庞加莱-本迪克森”理论。复杂程度最低的是纤维结，这里称为“深度零结”。这些结是相对罕见的，已经被经典地研究过，并且被相对地很好地理解了。下一个层次的复杂性是由深度1节表现出来的，它非常多（如David Gabai的论文所示），并提出了许多有趣的问题。深度一节目前是该项目的重点，已经获得的结果证实了他们的兴趣。其他需要研究的问题是紧叶理的光滑程度与拓扑结构的关系。虽然平滑性问题与3流形拓扑学家没有特别的关系，但一个强有力的例子可以证明它们与拓扑学的相关性是深刻的。在一个相当不同的方向上，提议者仍然对叶的遍历理论感兴趣，特别是异常最小集的遍历理论，以及哥德亿-维不变量是否检测到这样的最小集的相关问题（几乎可以肯定不是）。流形的叶状化是一种用低维块填充流形的方法。在余维为1的叶状情况下，这些块的维数比给定流形的维数小1。想想洋葱或洋蓟。流形的拓扑结构与它所支持的叶状结构类型密切相关，在熟练的操作中，这种关系已被塑造成研究流形拓扑结构的强大工具。研究流形拓扑的主要代数工具在高维流形的情况下效果最好，这是一个不直观的事实。因此，叶化所提供的几何工具在低维流形的情况下特别受欢迎。这方面的一个主要例子是对三维球体中结的补的研究，这是获得关于结本身信息的重要方法。

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.