Heegaard Floer homology for manifolds with torus boundary: properties and examples

Abstract

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:

Instanton L-spaces and splicing

“…We remark that Hanselman, Rasmussen, and Watson proved [HRW17] something more general than Theorem 1.1 in the Heegaard-Floer setting. In particular, they showed that the Heegaard-Floer homology of any toroidal manifold (not necessarily coming from a splice of knots in homology sphere L-spaces) has dimension at least five (over Z/2Z).…”

“…A Heegaard-Floer analogue of Theorem 1.1, with coefficients in Z/2Z rather than Q, was first established by Hedden and Levine in [HL16], and later generalized by Eftekhary [Eft15,Eft20] and then by Hanselman, Rasmussen, and Watson [HRW17]. Their approaches all used bordered Heegaard Floer homology or something similar.…”

The correspondence induced on the pillowcase by the earring tangle

“…Call a curve loop‐type (respectively, arc‐type ) if there exists a $$\delta >0$$ so that the curve is $$\delta$$‐loop‐type (respectively, $$\delta$$‐arc‐type). We remark that this terminology differs from the one in [40], where loop‐type curves are compact curves with the trivial local system.…”

“…In the torus boundary case, Hanselman, Rasmussen, and Watson interpreted the bordered Heegaard Floer type D structure invariant as an immersed curve $$\widehat{\mathit {HF}}(M)$$ in a once‐punctured torus [40]: $$\begin{equation*} (M, \partial M=T^2) \quad \mapsto \quad \widehat{\mathit {CFD}}(M)^{\mathcal {A}(T^2)} \quad \mapsto \quad \widehat{\mathit {HF}}(M) \looparrowright T^2\setminus 1\text{pt}. \end{equation*}$$Immersed curves in this context may contain several components, and each component comes with a local system.…”

“…The second is a classification result, stating that any twisted complex (up to homotopy equivalence) over the Fukaya category of a surface is equivalent to a collection of immersed curves in that surface, decorated with local systems. This was first proved by Haiden–Katzarkov–Kontsevich [39] by representation theoretic techniques, but a more geometric proof was given in [40], which provides a practical algorithm of obtaining an immersed curve $$\widehat{\mathit {HF}}(M)$$ from the twisted complex $$\widehat{\mathit {CFD}}(M)^{\mathcal {A}(F)}$$. Generalizations of this geometric approach to the fully wrapped case and fields other than characteristic 2 were obtained in [56].…”