The PI plans to study topology of 3-manifolds, especially Heegaard splittings of 3-manifolds. The first part of the project is to investigate the relation between the Heegaard genus of a 3-manifold and the rank of its fundamental group. This research is built on the PI's recent success in constructing a hyperbolic counterexample to the Rank versus Genus Conjecture. The second goal of the proposed research is to answer a long-standing question in 3-manifold topology concerning Heegaard genus and a degree-one map. This question asks whether it is possible to have a degree-one map from a 3-manifold to another 3-manifold with larger Heegaard genus. This question is related to the Poincare Conjecture as well as a few other important questions in 3-manifold topology. The third part of the project is to solve a conjecture of Morimoto and Moriah on tunnel number of knots in the 3-sphere. The PI plans to develop new tools and use techniques from his previous work 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 pair of simpler 3-dimensional pieces called handebodies along a 2-dimensional surface. This decomposition is called a Heegaard splitting. The PI plans to study 3-manifolds using Heegaard splittings. 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.
