Mathematical Sciences: Topology of Foliations and Foliated Knot Complements

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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.