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Filtered convergence and the scope of the Ozsváth–Szabó construction

Statement

Let $(C, d, F)$ be a filtered chain complex over a field with an increasing filtration $0 = F_{- 1} C \subseteq F_{0} C \subseteq \dots \subseteq F_{n} C = C$ and $d (F_{p} C) \subseteq F_{p} C$ . Its spectral sequence stabilizes and abuts to $H (C)$ with the image filtration, without any uniform bound outside the finite range $0 \leq p \leq n$ . If a particular filtered complex has been constructed from a fixed link diagram, this convergence assertion does not itself identify filtered complexes arising from different diagrams.

Statement references

Filtered chain complex convention

Bounded filtration convergence theorem

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, §1.1, “Further remarks and speculation,” printed pp. 3–4, especially p. 4

Weibel, An Introduction to Homological Algebra, Chapter 5

Lipshitz, Heegaard Floer Homologies, §5

Facts & Assumptions

Given: A filtered chain complex $(C, d, F)$ with $F_{- 1} C = 0$ and $F_{n} C = C$ for some finite $n$ .

[F1]

A filtered chain complex has $d (F_{p} C) \subseteq F_{p} C$ (Filtered chain complex).

[F2]

A finite filtration yields a spectral sequence abutting to homology with the image filtration (Bounded filtered complex spectral sequence abuts to filtered homology).

Proof

1.1F1given

The filtration has only the levels $0, \dots, n$ , so it is finite, exhaustive and separated; [F1] says each level is a subcomplex.

2.1F2step 1.1

Apply [F2] to this bounded filtered complex. Its pages stabilize and the limiting page is the associated graded of $H (C)$ with the filtration induced by the inclusions $F_{p} C ↪ C$ .

3.1F1F2step 2.1∎

This argument identifies the abutment for the fixed filtered complex only. No map between filtered complexes for two different link diagrams occurs in [F1] or [F2], so diagram-independent filtered quasi-isomorphism type is not a consequence of this convergence result.

Depends on

Used by

The Ozsváth–Szabó spectral sequence for a fixed link diagram Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ozsváth and Szabó, On the Heegaard Floer homology of branched double-covers, §1.1 Further remarks and speculation, printed pp. 3–4; filtered quasi-isomorphism discussion on p. 4 (standard reference, not scraped)
Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Lipshitz, Heegaard Floer Homologies, §5 (standard reference, not scraped)