Heegaard Floer homology and integer surgeries on links

TL;DR

The paper develops a comprehensive, combinatorial framework for computing Heegaard Floer homology of integral link surgeries via complete systems of hyperboxes. It introduces hyperbox and hypercube formalisms, the algebra of songs and symphonies, and a detailed treatment of polygon maps to manage higher compositions, all anchored by invariance under Heegaard moves and a grid-diagram approach. The authors demonstrate explicit computations for notable cases (e.g., Hopf link) and extend the machinery to mixed four-manifold invariants, connecting link Floer theory to four-dimensional topology. The result is a versatile, largely combinatorial pipeline for recovering $HF^-(Y_\ abla(L))$ and related cobordism maps from link data, with potential applications to link homologies and spectral sequences tied to surgery.

Abstract

Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) the link Floer homology of L and all its sublinks, together with several chain maps between these complexes. Further, we introduce a way of presenting closed four-manifolds with b_2^+ > 1 by four-colored framed links in the three-sphere. Given a link presentation of this kind for a four-manifold X, we then describe the Ozsvath-Szabo mixed invariants of X in terms of a complete system of hyperboxes for the link. Finally, we explain how a grid diagram produces a particular complete system of hyperboxes for the corresponding link.

Figures (56)

• Figure 1: An index zero/three link stabilization. The surface $\Sigma'$ is obtained from $\Sigma$ by deleting a disk and adding a cap, i.e., taking the connected sum with a sphere inside $Y$. This is the same picture as MOS, but we emphasize the fact that the construction is done inside the fixed 3-manifold $Y$, with a fixed link $L$.
• Figure 2: The positive Hopf link.
• Figure 3: A Heegaard diagram for the Hopf link. The thicker (red) curve is $\alpha$, while the thinner (black) curve is $\beta$.
• Figure 4: Another Heegaard diagram for the Hopf link. This picture is obtained from Figure \ref{['fig:hopf1']} by stabilizing twice and doing some handleslides. It has the advantage that the basepoints come in pairs $(w_1, z_1)$ and $(w_2, z_2)$, with $w_i$ and $z_i$ on each side of the curve $\beta_i$.
• Figure 5: The complex ${\mathcal{C}}^-({\mathcal{H}}, \Lambda)$ for $p_1 = p_2 = 2$. We show here the lattice $\mathbb{H}(L)$, as a union of various icons in the plane: black dots, white dots, and black diamonds. Each type of icon corresponds to a particular ${\operatorname{Spin^c}}$ structure on the surgered manifold. We also show how various parts of the differential ${\mathcal{D}}^-$ act on the lattice. Not shown is ${\mathcal{D}}^{00}$, which simply preserves each icon. Note that all parts of ${\mathcal{D}}^-$ preserve the type of the icon.

Theorems & Definitions (360)

• Theorem 1.1
• Theorem 1.2
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Remark 3.6
• Remark 3.7
• Remark 3.8