The PI plans to study the topology of 3-manifolds, especially Heegaard splitting of 3-manifolds, using methods of amalgamation, surgery, branched surface and lamination. The first goal of the project is to construct a hyperbolic counterexample to the Rank Conjecture. The second part of the proposed research is to use branched surfaces and laminations to study Dehn surgery on knots and links in reducible 3-manifolds. One goal is to prove a major part of the Cabling Conjecture. The third part of the project is to study various aspects of Heegaard splitting. There are many interesting connections between the proposed research and several active programs of other researchers. The PI plans to explore these connections and develop new tools to achieve these goals.
Three-manifolds are objects modeled on the 3-dimensional space that we are living in. A donut and the spatial universe are both examples of 3-manifolds. These objects arise naturally in many contexts in physical and other natural sciences and model many interesting phenomena. A geometric way of studying 3-manifolds is to cut a complicated 3-manifold into a collection of simpler 3-dimensional pieces along 2-dimensional surfaces. For example a Heegaard splitting is such a decomposition. Using this idea, one can also construct useful new 3-manifolds by doing surgery on well-understood ones. The PI plans to study 3-manifolds using Heegaard splitting and surgery. The research targets several central questions in low-dimensional topology and knot theory, which has potential impact on other areas of scientific investigations, such as the topological structures of DNA.
The main goal of this NSF funded project is to study low-dimensional topology. Topology is a branch of mathematics studying the qualitative shape and intrinsic properties of geometric objects. Topological ideas are present in almost all areas of today's mathematics. Manifolds are geometric objects that locally modeled on the Euclidean spaces. For instance, a 2-dimensional manifold is just a surface. The main focus of this project is 3-dimensional manifolds, often referred to as 3-manifolds. Three-manifold research has attracted significant attention in that 3-manifolds have beautiful geometric and combinatorial structures and many important questions remain unsolved. A better understanding of these structures is of central interest in geometry and topology and may have profound impact on physical and other natural sciences. The object that the PI has been studying is called Heegaard splitting. A Heegaard splitting is a simple and very useful decomposition of a 3-manifold along a 2-dimensional surface. Every 3-manifold has a Heegaard splitting. In this project, the PI has made significant progress in answering several important questions in 3-manifold topology using techniques he developed on Heegaard splittings. This work helps gain better understanding of 3-manifolds. A problem that the PI has solved in this project is a fundamental question asking whether or not the rank of the fundamental group equals the Heegaard genus for a hyperbolic 3-manifold. This question relates two fundamental invariants of a 3-manifold and was originally asked by Waldhausen in 1960s in an effort to prove the Poincare Conjecture. Another problem solved in this project is concerned with another type of surfaces in 3-manifolds, called incompressible surface. The PI has constructed a class of 3-manifolds that contain no incompressible surfaces but admit complicated Heegaard splittings. This construction is closely related to another important question in 3-manifold topology. The PI has also used techniques developed in this project to study various questions in knot theory and has successfully answered a conjecture of Morimoto and Moriah. The PI has used this NSF project to train two junior graduate students in summer reading and research programs at Boston College. The PI has also taught a graduate topic course and he has incorporated materials from this project into this course. The PI has given a mini-course during a workshop in Montreal using materials from this project. Most of the audiences in the mini-course were graduate students from various institutes in Canada and the US. The PI has also given lectures in conferences seminars on his research in this NSF project.
