[Paper Review] Heegaard Floer homology for manifolds with torus boundary: properties and examples

Jonathan Hanselman, Jacob Rasmussen+1 more|arXiv (Cornell University)|Oct 22, 2018

Geometric and Algebraic Topology5 citations

TL;DR

This paper establishes a geometric, curve-based description of Heegaard Floer homology for 3-manifolds with torus boundary using immersed curves and local systems in a punctured torus. It proves key invariance properties under Spin^c conjugation, relates the invariant to knot Floer homology and the Thurston norm, and provides a geometric interpretation of the gradings in bordered Floer homology, with applications to genus one mutation invariance and splicing constructions.

ABSTRACT

This is a companion paper to earlier work of the authors, 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 $HF^\pm$. Many examples are included.

Motivation & Objective

• To provide a geometric, curve-based invariant for 3-manifolds with torus boundary using immersed curves and local systems in the punctured torus.
• To establish a connection between this invariant and classical invariants such as the Thurston norm, knot Floer homology, and Turaev torsion.
• To give a geometric interpretation of the gradings in bordered Heegaard Floer homology and prove symmetry under Spin^c conjugation.
• To demonstrate genus one mutation invariance in Heegaard Floer homology for closed 3-manifolds using this framework.
• To explore speculative connections with Seiberg-Witten theory, Khovanov homology, and HF^+ via the curve invariant.

Proposed method

• Use the bordered Heegaard Floer homology framework to define an invariant $\widehat{\mathit{HF}}(M)$ as a collection of immersed curves with local systems in $\partial M \setminus \{z\}$, up to regular homotopy and isomorphism.
• Apply the structure theorem for type D structures to realize $\widehat{\mathit{HF}}(M)$ as a geometric object, with a constructive algorithm implemented computationally.
• Compute $\widehat{\mathit{HF}}(Y)$ for closed 3-manifolds $Y = M_0 \cup_h M_1$ via immersed Lagrangian intersection Floer homology of $\widehat{\mathit{HF}}(M_0)$ and $h(\widehat{\mathit{HF}}(M_1))$ in the boundary torus.
• Interpret the Spin^c grading package geometrically via the action of the mapping class group on the curve configuration.
• Use the curve invariant to compute $\mathit{HF}^+$ via a mapping cone construction, analyzing bigons and local systems in the intersection complex.
• Analyze examples including spliced trefoils and rational homology spheres to test invariance and detect obstructions to $\mathit{HF}^+$ reconstruction from $\widehat{\mathit{HF}}$ alone.

Experimental results

Research questions

• RQ1 How can the Heegaard Floer homology of a 3-manifold with torus boundary be geometrically described using curves and local systems?
• RQ2 What is the relationship between this curve invariant and classical invariants like the Thurston norm and knot Floer homology?
• RQ3 Does the geometric grading structure in bordered Floer homology exhibit symmetry under Spin^c conjugation, and what are its topological consequences?
• RQ4 To what extent does $\widehat{\mathit{HF}}(M)$ determine $\mathit{HF}^+(M_0 \cup_h M_1)$ for spliced manifolds?
• RQ5 Under what conditions is $\mathit{HF}^+(M_0 \cup_h M_1)$ isomorphic to the $\mathit{HF}^+$ of the curve intersection $\mathit{HF}^+(\widehat{\mathit{HF}}(M_0), h(\widehat{\mathit{HF}}(M_1)))$?

Key findings

• The invariant $\widehat{\mathit{HF}}(M)$ for a 3-manifold with torus boundary is fully determined by a collection of immersed curves with local systems in the punctured torus, up to regular homotopy and isomorphism.
• The dimension of $\widehat{\mathit{HF}}(Y)$ for a closed 3-manifold $Y = M_0 \cup_h M_1$ is computed as the minimal intersection number between $\widehat{\mathit{HF}}(M_0)$ and $h(\widehat{\mathit{HF}}(M_1))$, with $\dim \widehat{\mathit{HF}}(Y) = 5$ for the spliced trefoil example.
• The geometric grading package in bordered Floer homology is shown to be symmetric under Spin^c conjugation, leading to genus one mutation invariance in Heegaard Floer homology.
• For the spliced trefoil, $\mathit{HF}^+(Y) \cong \mathcal{T}_+ \oplus \mathbb{F}^3$, computed via 7 generators and 8 bigons in the intersection complex.
• There exist examples where $\mathit{HF}^+$ cannot be reconstructed from $\widehat{\mathit{HF}}$ alone, particularly when local systems have multiplicity greater than one, indicating a need for refined invariants.
• The paper conjectures that $\mathit{HF}^+(M_0 \cup_h M_1) \cong \mathit{HF}^+(\widehat{\mathit{HF}}(M_0), h(\widehat{\mathit{HF}}(M_1)))$ when $M_0$ and $M_1$ are Floer simple or have good, multiplicity-one local systems.