Low-Dimensional Topology via Bordered Floer Theory

Princeton University, Princeton NJ

Investigators

Abstract

This research project in low-dimensional topology investigates the global shape of various three-dimensional spaces and of knotted curves and surfaces within them. Since the space we inhabit is three-dimensional, this field has wide-ranging applications, from understanding the possible shapes of the universe to describing the knotting of polymers and DNA molecules. Understanding the properties of three-dimensional spaces requires an array of sophisticated tools from a range of disciplines within mathematics, including algebraic topology, geometry, analysis, and representation theory. Each tool for studying three-dimensional spaces carries with it a notion of which spaces are "simple" and which are "complicated." The primary goal of this project is to explore the relationship between different tools used to describe three-dimensional spaces and the corresponding notions of simplicity. Ultimately the investigator hopes to consolidate and deepen our knowledge of these spaces and to build bridges between a variety of mathematical fields. A key technical tool to be used in this investigation is bordered Heegaard Floer homology, a version of Heegaard Floer homology for 3-manifolds with boundary. In particular, the project centers on developing a new geometric interpretation of bordered Heegaard Floer invariants; this is a concrete realization of a deep connection between bordered Heegaard Floer homology and certain Fukaya categories. In the case of torus boundary, the invariants may be interpreted as decorated immersed curves in the boundary torus. This framework greatly simplifies computations and leads to proofs of interesting gluing results. This will be applied to the classification of L-spaces (3-manifolds that are "simple" with respect to Heegaard Floer homology). As one application, the investigator seeks to confirm a conjecture equating three measures of simplicity for 3-manifolds: being an L-space, having non-left-orderable fundamental group, and not admitting a co-orientable taut foliation. Other goals of the project include restricting the possible decompositions of L-space knots and relating Heegaard Floer homology to the complexity of the Jaco-Shalen-Johannson decomposition.