Higher Structure in Low-Dimensional Floer Theories

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Humanity has been interested in the geometry of three-dimensional spaces since long before recorded history; since Einstein's work on general relativity, the geometry of four-dimensional spaces has also been of central importance, with applications now as ubiquitous as the global positioning system. Mathematicians have divided the study of spaces into two parts: qualitative questions (like whether a space has a hole in it), which is the field of topology, and quantitative questions (like the diameter of the hole), which is the field of geometry. Most, though not all, applications require quantitative answers, but before one can start on quantitative answers---the geometry---one needs to understand the qualitative behavior---low-dimensional topology, the focus of this National Science Foundation funded project. Indeed, in four-dimensions, even many of the most basic topological questions remain unanswered---questions about how many different simple spaces there are, how spaces can be cut up into simpler pieces, and what kinds of surfaces one can fit in given four-dimensional spaces. By using relations with other parts of mathematics and high-energy physics, this proposal seeks to advance human understanding by developing tools to answer some of these basic questions. This project, which focuses on applications of symplectic topology, algebraic topology, and representation theory to low-dimensional topology, has four interrelated goals. The first goal is to extend Lipshitz-Ozsváth-Thurston's "bordered Heegaard Floer homology" invariant of three-manifolds with boundary from the "hat" variant to the richer "minus" variant, for three-manifolds with torus boundary. The second goal is to prove a higher naturality result for Heegaard Floer homology, giving a functor from an appropriate quasi-category of decorated cobordisms to the quasi-category of chain complexes. The third goal is to prove new localization results relating the Heegaard Floer homology of a space with the Heegaard Floer homology of its branched covers. The fourth is to extend Lipshitz-Sarkar's stable homotopy refinement of Khovanov homology to cobordisms between knots and to other variants of Khovanov homology, including Lee homology. These tools are expected to have applications to concordance, homology cobordism, and other problems at the interface of three- and four-dimensional topology.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.