LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The tau level is preserved by filtered chain homotopy equivalence

Statement

Let $C$ and $D$ be finite integer-filtered chain complexes over $F_{2}$ , each with total homology isomorphic to $F_{2}$ . A filtered chain homotopy equivalence consists of filtration-preserving chain maps $f : C \to D$ and $g : D \to C$ such that $g f$ and $f g$ are chain homotopic to the respective identity maps by homotopies that preserve every filtration level. Then $τ (C) = τ (D)$ .

Statement references

The first filtration level whose inclusion has nonzero image in total homology

Ozsvath and Szabo, Knot Floer homology and the four-ball genus, §2.1

Hom, A survey on Heegaard Floer homology and concordance, §3.1

Facts & Assumptions

Given: $C, D$ and filtered chain maps $f, g$ satisfying the filtered chain homotopy equivalence hypotheses in the Statement.

[F1]

The Alexander filtration and tau level: for a finite filtered complex with one-dimensional total homology, $τ (C)$ is the least integer $s$ for which $H (F_{s} C) \to H (C)$ has nonzero image.

Proof

1.1F1givenalgebra

Put $t = τ (C)$ . By [F1], some class in $H (F_{t} C)$ maps to a nonzero class in $H (C)$ . Since $f$ preserves filtration, its restriction sends $F_{t} C$ to $F_{t} D$ . The square formed by these restricted maps and the inclusions into total complexes commutes, so the induced map sends that class to its image under $f_{*}$ . The map $f_{*}$ is an isomorphism because $g_{*}$ is its homotopy inverse; hence the resulting class in $H (D)$ is nonzero. Therefore $H (F_{t} D) \to H (D)$ has nonzero image and $τ (D) \leq t$ .

2.1F1step 1.1algebra∎

The same argument using the filtration-preserving map $g$ gives $τ (C) \leq τ (D)$ . Together with step 1.1 this yields $τ (C) = τ (D)$ .

Depends on

The Alexander filtration and tau level

Used by

The tau level is independent of the normalized Heegaard model Lemma

Dependency tree · two levels

5 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

Peter S. Ozsvath and Zoltan Szabo, Knot Floer homology and the four-ball genus (standard reference, not scraped)
Jennifer Hom, A survey on Heegaard Floer homology and concordance (standard reference, not scraped)