Heegaard splitting and surgery of 3-manifolds

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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.