Topics in low-dimensional topology

Boston College, Chestnut Hill MA

Investigators

Abstract

The PI plans to explore the interconnections between geometric, topological and combinatorial structures of 3-manifolds through efficient triangulations, branched surfaces and Heegaard splittings. The first part of the project is to construct geometric triangulations for certain 3-manifolds. The second part of the 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 the Cabling Conjecture and to give a new proof of the Property R theorem for knots. The third part of the project is to study various aspects of Heegaard splittings. There are many interesting connections between the research and several major questions in low-dimensional topology. Results from this project may pave the road toward the resolution of some difficult but important conjectures in low-dimensional topology. The PI plans to develop new tools and use techniques from his previous work, such as branched surfaces a nd laminations, 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. Three-manifolds have beautiful geometric and topological properties and these properties are encoded in certain combinatorial information. The PI will explore such combinatorial information through some geometric objects called efficient triangulation, branched surface and Heegaard splitting. 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.