This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure CFD(M ) may be viewed as a set of immersed curves decorated with local systems in ∂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 between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of 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 HF . In particular, it follows that a prime rational homology sphere Y with 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 characterisation 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.

We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. Using the results of these two computations, we describe an algorithm for computing HF-hat of any graph manifold.Comment: 59 pages, 21 figures, new version corrects typos and adds a short discussion of grading

This is a companion paper to earlier work of the authors (Preprint, arXiv:1604.03466, 2016, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We establish a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under Spin 𝑐 conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds.Finally, we include more speculative discussions on relationships with Seiberg-Witten theory, Khovanov homology, and 𝐻𝐹 ± . Many examples are included.M S C 2 0 2 0 57K31, 57K18, 57R58 (primary) Bordered Heegaard Floer homology provides a toolkit for studying the Heegaard Floer homology of a three-manifold 𝑌 decomposed along a surface. This theory was introduced and developed by Lipshitz, Ozsváth, and Thurston [29], and has been studied in some detail in the case of essential tori as these are relevant to questions related to the JSJ decomposition of 𝑌. In the authors' previous work [12], a geometric interpretation of the bordered Heegaard Floer homology of a three-manifold with torus boundary 𝑀 is established. In particular, we proposed:

This is a companion paper to earlier work of the authors [10], which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under Spin c conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds. Finally, we include more speculative discussions on relationships with Seiberg-Witten theory, Khovanov homology, and HF ± . Many examples are included.

If Y is a closed orientable graph manifold, we show that Y admits a coorientable taut foliation if and only if Y is not an L-space. Combined with previous work of Boyer and Clay, this implies that Y is an L-space if and only if π1(Y ) is not left-orderable.