Mathematical Sciences: Structure of Codimension one Foliations

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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.