Bordered Floer homology for manifolds with torus boundary via immersed curves

Jonathan Hanselman, Jacob Rasmussen, Liam Watson

Published 2016-04-12Version 1

We give a geometric interpretation of bordered Floer homology for a natural class of three-manifolds with torus boundary; namely, the loop-type manifolds defined by the first and third authors. From this viewpoint, the invariant associated with a loop-type manifold is a collection of immersed curves in the punctured torus, and the pairing theorem of Lipshitz, Ozsvath and Thurston amounts to taking the intersection Floer homology of the corresponding curves. Using these ideas, we are able to strengthen our results with S. Rasmussen on when an L-space can contain an incompressible torus while simultaneously simplifying the proofs. Other applications include a dimension inequality for Heegaard Floer homology under pinching and a new description of the interval of L-space slopes of a Floer simple manifold.