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The Ozsváth–Szabó spectral sequence for a fixed link diagram

Statement

Assume AC and the ACω Heegaard Floer analytic package. Let $D$ be any finite regular diagram of a nonempty oriented link $L \subset S^{3}$ , with a marked point away from crossings. There is a spectral sequence over $F_{2}$ associated to the link-surgery filtered complex for this fixed diagram whose $E_{2}$ page is isomorphic, with the normalized grading shifts of the chosen smoothing convention, to $\tilde{K h} (\overline{L}, p; F_{2})$ computed from the mirror diagram, and whose abutment is $\hat{H F} (Σ (L); F_{2})$ with its image filtration. This is a fixed-diagram construction; it does not assert a diagram-independent filtered quasi-isomorphism type.

Statement references

Branched cover convention

Filtered link-surgery module

E1 differential identification

Finite filtration convergence

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, Theorem 6.3 and proof

Lipshitz, Heegaard Floer Homologies, §5.3

Facts & Assumptions

Given: AC, a nonempty oriented link diagram $D$ with a basepoint, and the stated ACω Floer disk and polygon theory.

[F1]

The diagram determines the surgery cube of branched covers and its filtration (Link-surgery filtered complex for a double branched cover).

[F2]

The surgery cube is a filtered chain complex with total homology identified with the hat Floer homology of the branched cover (Ozsváth–Szabó link-surgery filtered complex and exact triangle, Hat Heegaard Floer homology is a closed three-manifold invariant).

[F3]

The first page and its differential identify with the reduced Khovanov cube of the mirror diagram (The E1 differential of the link-surgery complex is reduced Khovanov).

[F4]

A finite filtration yields a convergent spectral sequence to the associated graded of the filtered homology (Filtered convergence and the scope of the Ozsváth–Szabó construction).

Proof

1.1F1F2given

The crossing resolutions of $D$ give the vertices of the cube, and the surgery maps give the differential components. By [F2] these components square to zero and preserve the increasing cube filtration, so the associated spectral sequence is defined.

2.1F2F4step 1.1

The filtration has only the $n + 1$ levels $0, \dots, n$ , where $n$ is the number of crossings. By [F4] the spectral sequence stabilizes and abuts to the image filtration on the total homology, which [F2] identifies with $\hat{H F} (Σ (L); F_{2})$ .

2.2F3step 1.1

The associated graded vertex groups form the $E_{1}$ page, and their first differential is the sum of the single-crossing edge maps. By [F3], this complex is isomorphic to the reduced Khovanov complex of the mirror diagram; taking its cohomology gives the stated $E_{2}$ page.

3.1F2F3F4step 1.1step 2.1step 2.2∎

The proof used one fixed $D$ and the finite filtration only. It supplies no filtered map between the constructions for two diagrams, so it establishes the stated fixed-diagram spectral sequence but no diagram-independent filtered quasi-isomorphism. Here [F2] uses the explicitly external OS surgery construction and filling comparison, and [F3] uses its actual compatible unlink maps with the proved finite reduced-algebra binding. Those imports retain their stated scope; no missing geometric map is inferred from finite convergence.

Depends on

Used by

The Khovanov to branched-cover Heegaard Floer spectral sequence Theorem

Dependency tree · two levels

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Sources

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, Theorem 6.3 and proof (standard reference, not scraped)
Lipshitz, Heegaard Floer Homologies, §5.3 (standard reference, not scraped)