[Paper Review] On the Heegaard Floer homology of branched double-covers

Peter Ozsváth, Zoltán Szabó|ArXiv.org|Sep 9, 2003

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper establishes a spectral sequence from the reduced Khovanov homology of the mirror of a link $L \subset S^3$ to the Heegaard Floer homology of its branched double cover $\Sigma(L)$, with the $E^2$ term isomorphic to Khovanov's homology and the $E^\infty$ term isomorphic to $\widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z})$. For alternating links, $\widehat{HF}(\Sigma(L))$ is completely determined by the determinant of $L$, and $\Sigma(L)$ is an $L$-space, confirming a strong link between quantum invariants and 3-manifold topology.

ABSTRACT

Let $L\subset S^3$ be a link. We study the Heegaard Floer homology of the branched double-cover $Σ(L)$ of $S^3$, branched along $L$. When $L$ is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose $E^2$ term is a suitable variant of Khovanov's homology for the link $L$, converging to the Heegaard Floer homology of $Σ(L)$.

Motivation & Objective

• To establish a deep connection between quantum link invariants (Khovanov homology) and 3-manifold invariants (Heegaard Floer homology) via branched double covers.
• To compute $\widehat{HF}(\Sigma(L))$ for alternating links, showing it is determined solely by the determinant of $L$.
• To construct a spectral sequence linking the reduced Khovanov homology of the mirror of $L$ to $\widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z})$, providing a bridge between quantum and gauge-theoretic invariants.
• To extend the skein exact sequence for $\widehat{HF}(\Sigma(L))$ to a full spectral sequence, generalizing the alternating case.

Proposed method

• Use a link surgeries spectral sequence in Heegaard Floer homology, constructed via a filtered complex indexed by subsets of crossings in a link diagram.
• Define a cube of resolutions $\mathcal{D}(I)$ for a link diagram $\mathcal{D}$, where each $I \subset \{0,1\}^\ell$ corresponds to a resolution of crossings.
• Construct a canonical identification between the reduced Khovanov complex $\widetilde{CKh}(\mathcal{D}(L), m)$ and the $E^1$-page of the spectral sequence via homology classes in $H_1(\Sigma(\mathcal{D}(I)))$.
• Prove commutativity of a diagram relating the differential in the Khovanov complex to the differential in the Heegaard Floer complex via cobordism-induced maps.
• Apply a standard 'flattening' procedure to convert the cube filtration into a $\mathbb{Z}$-filtration, yielding a $\mathbb{Z}$-filtered spectral sequence.
• Use the resulting spectral sequence to derive inequalities between the rank of $\widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z})$ and the reduced Khovanov homology rank, with the determinant of $L$ as a lower bound.

Experimental results

Research questions

• RQ1 Can the Heegaard Floer homology of the branched double cover $\Sigma(L)$ be computed from quantum invariants of $L$?
• RQ2 Does a spectral sequence exist that connects reduced Khovanov homology of the mirror of $L$ to $\widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z})$?
• RQ3 For alternating links, is $\widehat{HF}(\Sigma(L))$ completely determined by the determinant of $L$?
• RQ4 How do the $\mathrm{Spin}^c$ structures and $\mathbb{Q}$-grading on $\widehat{HF}(\Sigma(L))$ relate to the diagram of $L$?
• RQ5 What constraints do the $\mathrm{Spin}^c$-graded and filtered structure of $\widehat{HF}(\Sigma(L))$ impose on the intersection forms of negative-definite four-manifolds bounding $\Sigma(L)$?

Key findings

• There exists a spectral sequence whose $E^2$ term is the reduced Khovanov homology of the mirror of $L$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients, and which converges to $\widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z})$, establishing a direct link between quantum and 3-manifold invariants.
• For any alternating link $L$, $\widehat{HF}(\Sigma(L))$ is an $L$-space, and its rank equals the order of $H^2(\Sigma(L); \mathbb{Z})$, i.e., the determinant of $L$, showing that $\Sigma(L)$ is an ungraded Heegaard Floer homology lens space.
• The rank of $\widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z})$ satisfies the inequality $\det(L) \leq \mathrm{rk}_{\mathbb{Z}/2\mathbb{Z}} \widehat{HF}(\Sigma(L); \mathbb{Z}/2\mathbb{Z}) \leq \mathrm{rk}_{\mathbb{Z}/2\mathbb{Z}} \widetilde{\mathrm{Kh}}(L)$, with equality in the first inequality if and only if $\Sigma(L)$ is an $L$-space.
• The $\mathrm{Spin}^c$ structures on $\Sigma(L)$ are explicitly determined by the alternating diagram of $L$, and the $\mathbb{Q}$-grading on $\widehat{HF}(\Sigma(L))$ is fully computable from the diagram.
• The spectral sequence is constructed via a filtered complex indexed by resolutions of crossings, with the $E^1$-page isomorphic to the reduced Khovanov complex, and the differentials induced by cobordisms between branched double covers of resolutions.
• The construction generalizes the skein exact sequence for $\widehat{HF}(\Sigma(L))$ to a full spectral sequence, providing a computational framework for $\widehat{HF}(\Sigma(L))$ beyond the alternating case.