Three-manifolds are objects modeled on the three-dimensional space that we are living in. A doughnut and the spatial universe are both examples of three-manifolds. These objects arise naturally in many contexts in physical and other natural sciences and model many interesting phenomena. A geometric way of studying them is to cut a complicated three-manifold into simpler three-dimensional pieces along two-dimensional surfaces. For example, a so-called Heegaard splitting is such a decomposition. One can also use a technique known as surgery on a three-manifold that replaces a geometric piece with a different one. In this project, the PI plans to study various questions on three-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.
In this project, the PI plans to study topology of three-manifolds. The first part of the research is to study a long-standing conjecture concerning Heegaard genus and degree-one map. The conjecture can be translated into a study of how Heegaard genus changes under a special type of surgeries and the PI plans to study such surgeries. The PI also plans to study several related questions on Heegaard genus of certain amalgamated three-manifolds. The second part of the project is to explore a new approach to proving the Berge Conjecture. This approach is inspired by the PI's observation that if there is a certain intersection pattern, then one can perform a stabilization and then an unstabilization, changing a Heegaard splitting of a knot exterior (in a lens space) into a meridionally primitive splitting. The PI plans to develop new tools and use techniques from his previous work to achieve these goals.
