Mathematical Sciences: Structure of Codimension one Foliations

该项目继续与Lawrence W.Conlon(华盛顿大学圣路易斯分校)共同研究协维一叶的结构，这项工作始于1975年。他们已经证明，每个开放的表面都可以作为具有某些叶理的叶子出现，并询问什么样的准等距表面可以作为叶子出现。马尔可夫例外局部极小集是良好的，问题是每个例外极小集是否在某种意义上本质上是马尔可夫集，或者，如果不是，至少具有马尔可夫集的许多性质。在一系列引人注目的论文中，加拜研究了纽结补语的叶结构，但在这一领域仍有许多工作要做。例如，首席研究人员打算构造更容易想象的纽结补体的叶，然后使用某个第二同调群的瑟斯顿球来描述某些纽结补体的所有紧的叶层。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 continues joint research with Lawrence W. Conlon (Washington University in St. Louis) on the structure of codimension-one foliations, work begun in 1975. They have shown that every open surface can occur as a leaf of some foliation and ask what quasi-isometry types of surfaces can occur as leaves. Markov-exceptional local minimal sets are well-behaved, and the question arises whether every exceptional minimal set is in some sense essentially a Markov one or, if not, at least has many of the properties of a Markov one. In a remarkable series of papers, Gabai has studied foliations of knot-complements, but much remains to be done in this area. For example, the principal investigator intends to construct foliations of knot-complements that are easier to visualize and then to use the Thurston ball of a certain second homology group to describe all taut foliations of certain knot- complements. 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.