Heegaard Splitting and Topology of 3-Manifolds

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Three-manifolds are objects modeled on the three-dimensional space that we live in. A donut and the spatial universe are both examples of three-manifolds. These objects arise naturally in many contexts in physical and other natural sciences, and can be used to model many interesting phenomena. The main goal of this project is to study the mathematical properties of three-manifolds. The PI plans to investigate some central questions in a branch of mathematics, known as low dimensional topology. These questions are concerned with how three-manifolds change under certain maps as well as operations called surgeries. The major tool that the PI uses is a topological structure called Heegaard splitting, which is a decomposition of a complicated three-manifold into simpler pieces along a two-dimensional surface. This research targets some of the fundamental questions in low-dimensional topology and knot theory. It also has a potential impact on other areas of scientific investigations, such as the topological structures of DNA. In this project, the PI will study the topology of three-manifolds. The project has three major parts. The first part is to explore a new approach to proving the Berge Conjecture. The Berge Conjecture can be divided into two halves: the first half is to prove the Berge Conjecture for knots with tunnel number one, and the second half is to show that if a nontrivial knot in the three-sphere admits a lens-space Dehn surgery, then the knot must have tunnel number one. The PI and his collaborators have carried out an in-depth study on the first half. The same approach may lead to a proof of the second half of the Berge Conjecture. The second part of the research is to study a long-standing conjecture concerning Heegaard genus and degree-one map. The PI will investigate this conjecture using special type of surgeries. The objective of the last part of the research is to study several fundamental questions in three-manifold topology concerning Heegaard splittings and curve complex. The PI plans to develop new tools and use techniques from his previous work to achieve these goals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.