Title for Jonathan Hanselman: "Heegaard Floer invariants for manifolds with torus boundary via immersed curves"

 

Abstract:  Heegaard Floer homology is an important topological invariant of closed 3-dimensional manifolds. I will describe a relative version of this invariant in the case of torus boundary: 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 decorated immersed curves in the torus—so that when two such manifolds are glued the Heegaard Floer homology of the resulting 3-manifold is recovered from the intersection of the corresponding curves. These curves are a reformulation of the bordered Heegaard Floer invariants 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 nice applications, including a rank inequality for Heegaard Floer homology of toroidal manifolds and invariance of Heegaard Floer homology under genus one mutation. This is joint work with J. Rasmussen and L. Watson.