Bordered Floer homology for manifolds with torus boundary via immersed curves

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M M is such a manifold, we show that the type D structure C F D ^ ( M ) \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in ∂ M \partial M . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of H F ^ \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of H F ^ \widehat {HF} . In particular, it follows that a prime rational homology sphere Y Y with H F ^ ( Y ) > 5 \widehat {HF}(Y)>5 must be geometric. Other results include a new proof of Eftekhary’s theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.