The PI will investigate various problems in three-dimensional topology. Specific topics that will be addressed are: (1) Exceptional Dehn Filling, (2) Bridge Number, Heegaard Genus and Dehn Surgery, and (3) L-Spaces and Left-Orderability. The context of (1) is the desire to classify all hyperbolic three-manifolds with two non-hyperbolic Dehn fillings; the present emphasis will be on the case where one of the fillings is Seifert fibered and the distance between the fillings is at least 5. Project (2) will consider manifolds obtained by Dehn surgery on knots in the three-sphere, the aim being to find upper bounds on the Heegaard genus of the manifold and the bridge number of the dual knot. The focus of (3) is the conjecture that a prime rational homology three-sphere is an L-space if and only of its fundamental group is not left-orderable.
Three-dimensional topology is the study of the large-scale structure of three-manifolds, abstract spaces that are locally like the ordinary space in which we live. As a result of Perelman's proof of the Geometrization Conjecture about ten years ago, we now have a very good picture of the qualitative nature of such spaces. Nevertheless, three-dimensional topology is extremely rich, with many different kinds of mathematics playing a role, and the proposed research will seek to further our understanding of various aspects of the subject. Specifically, (1) will study when the geometry of a three-manifold degenerates on having a certain simple space attached to it, (2) is part of the exploration of the fundamental relationship between three-manifolds and knots, and (3) will investigate a mysterious connection that appears to exist between two very different aspects of three-dimensional topology, one analytical, the other algebraic.
