Topology Seminar: Jonathan Hanselman

To a 3-manifold with torus boundary, we can associate an element of the Fukaya category of the punctured torus—that is, a collection of immersed curves in the torus, decorated with local systems—such that when two such manifolds are glued the Heegaard Floer homology of resulting 3-manifold is recovered from Floer homology of the corresponding curves. These curves are a reformulation of the bordered Heegaard Floer defined by Lipshitz, Ozsvath, and Thurston, but their geometric nature makes them more user friendly. We will discuss some properties of these curves and some applications, including invariance of Heegaard Floer homology under genus one mutation and a rank inequality for Heegaard Floer homology of toroidal manifolds. This is joint work with J. Rasmussen and L. Watson.