Bordered manifolds with torus boundary and the link surgery formula

TL;DR

The paper develops a bordered Heegaard Floer framework for 3-manifolds with torus boundary by recasting Manolescu–Ozsváth’s link surgery formula as type-$D$ modules over the knot-surgery algebra $\mathcal{K}$ and its multi-component link algebra $\mathcal{L}_{\ell}$. It proves a connected-sum (gluing) formula realized as an $A_\infty$-tensor product over $\mathcal{K}$, enabling the computation of CF$^-$ for manifolds obtained by gluing along torus boundaries and extending to splices via a torus-diffeomorphism bimodule. The work then develops a rich algebraic framework—encompassing $A_\infty$-modules, DA-bimodules, hypercubes, and linear topological spaces—to model the link surgery data, with concrete examples (Hopf link, rational solid tori) and interpretations of dual-knot and splicing phenomena in terms of minimal models and box-tensor products. Substantial attention is given to arc-system dependence, generalized pair-of-pants bimodules, and the invariant-theoretic aspects of bordered modules, laying groundwork for verifying lattice-homology conjectures and enabling new computations in 3-manifold HF$^-$. The constructions yield a scalable, algebraic toolkit for computing Heegaard Floer invariants of broad classes of 3-manifolds through surgery-like operations on bordered pieces.

Abstract

In this paper, we develop a theory of bordered $\mathit{HF}^-$ using the link surgery formula of Manolescu and Ozsváth. We interpret their link surgery complexes as type-$D$ modules over an associative algebra $\mathcal{K}$, which we introduce. We prove a connected sum formula, which we interpret as an $A_\infty$-tensor product over our algebra $\mathcal{K}$. Topologically, this connected sum formula may be viewed as a formula for gluing along torus boundary components. We compute several important examples. We show that the dual knot formula of Hedden--Levine and Eftekhary may be interpreted as the $DA$-bimodule for a particular diffeomorphism of the torus. As another example, if $K_1$ and $K_2$ are knots in $S^3$, and $Y$ is obtained by gluing the complements of $K_1$ and $K_2$ together using an orientation reversing diffeomorphism of their boundaries, then our theory may be used to compute $\mathit{CF}^-(Y)$ from $\mathit{CFK}^\infty(K_1)$ and $\mathit{CFK}^\infty(K_2)$. We additionally compute the type-$D$ modules for rationally framed solid tori. Our theory also computes the Heegaard Floer homology of all 3-manifolds which bound a plumbing of a tree of disk bundles over 2-spheres. In a subsequent article, we use this work to verify Némethi's conjecture about lattice homology.

Figures (33)

• Figure 1.1: The type-$D$ module for the $0$-framed trefoil complement. The superscript indicates the idempotent.
• Figure 1.2: Left: A bordered Heegaard diagram for a solid torus in the Lipshitz-Ozsváth-Thurston theory. Right: A Heegaard diagram of an unknot with a beta parallel arc system $\mathscr{A}$ (passing through the alpha curve). In our theory, the corresponding bordered manifold would be represented by the module for an unknot (whose complement is a solid torus). The choice of alpha arcs on the boundary of the bordered Heegaard diagram on the left is analogous to the choice of a beta parallel arc system on the right.
• Figure 3.1: Equalities of trees in $M_{n,*}^{\mathrm{red}}$.
• Figure 3.2: An example of a trees contributing to $L\Gamma_9(\Psi_9)$.
• Figure 4.1: The maps appearing in the homological perturbation lemma for $A_\infty$-modules.

Theorems & Definitions (325)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9
• Conjecture 1.10