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Ni's sutured Floer and Gabai product criterion

Statement

Assume full AC. Let $K$ be a null-homologous knot in a closed connected oriented $3$ -manifold $Y$ , let $Y ∖ K$ be irreducible, and let $F$ be a genus- $g$ Seifert surface. If the integral top Alexander group $\hat{H F K} (Y, K, [F], g; Z)$ is isomorphic to $Z$ , then the sutured manifold obtained by cutting the compact knot exterior along $F$ is a product sutured manifold. The argument uses Ni's homology-product reduction, the horizontal and separating product-annulus decompositions, the characteristic product region, and Gabai's taut-foliation input under their stated hypotheses.

Statement references

The integral top Alexander group and normalization

Product sutured manifold and characteristic product region

Sutured Floer homology notation and coefficients

Ni, Knot Floer homology detects fibred knots, §§3–7

Juhasz, Floer homology and surface decompositions, §§8–9

Gabai, Foliations and the topology of 3-manifolds III

Facts & Assumptions

Given: Full AC, the knot, exterior, surface and integral group in the Statement. A Seifert surface is connected, compact, oriented and properly embedded in the compact exterior, with its one boundary component the preferred longitude. The Axiom of Choice supplies the full Choice assumption.

[F1]

The resolved signed top/outer comparison in Analytic inputs for Ni's sutured knot Floer decompositions identifies the actual grading- $g$ complex with the actual $F$ -cut complex, with its induced coherent orientation class. Thus it identifies their integral homologies. This is also the exact external Theorem 1.5 of Juhász, Floer homology and surface decompositions, published p.300, for a null-homologous knot in any closed connected oriented $Y$ and any Seifert surface $F$ ; no three-sphere or rational-homology-sphere hypothesis occurs.

[F2]

External vanishing input: Juhász, Holomorphic discs and sutured manifolds, Proposition 9.18, published pp.1452–1453: if a balanced sutured manifold is irreducible and not taut, its integral sutured Floer homology is zero. This is used by contrapositive, independently of the resolved taut-to-nonzero hierarchy theorem.

[F3]

External nonproduct input: Juhász, Floer homology and surface decompositions, Theorem 9.7, published pp.343–348: every taut balanced sutured manifold which is not a product has integral sutured Floer homology containing a copy of $Z^{2}$ . Its geometric reduction, including the characteristic-pair step, is an external input here, not a locally reconstructed consequence of the $S^{3}$ JSJ theorem. The source's proof reduces by horizontal/product-annulus operations and produces two taut decompositions with disjoint outer ${Spin}^{c}$ sets. Their nonzero summands and signed cut maps agree with the scopes of the preserved local hierarchy and analytic suppliers. In particular no integral homology tensor formula without Tor is assumed.

[F4]

Corrected external characteristic-pair input: the original Ni characteristic-product-region citation is not imported uncorrected. Ni's Corrigendum, author-version pp.1–3, Theorem 4 and Definitions 5–6, uses a characteristic Seifert pair for a Haken manifold pair and a maximal gapless product pair containing its characteristic product pair. Theorem 6.2-prime assumes irreducible balanced, connected suture, vertically prime, and $\hat{H F S} ≅ Z$ ; it says that product-annulus homologies span $H_{1} (M)$ . Corollary 7 says that the characteristic product pair then surjects onto $H_{1} (M)$ . The corrected main proof on p.3 supplies the product reduction in the original arbitrary- $Y$ setting. These are source qualifications for the external product argument in F3 and Ni's Theorem 1.1, not assertions of the false universal ambient enclosure used in the uncorrected paper. None is deduced from the retained $S^{3}$ knot-exterior JSJ Statement.

Proof

technique · direct, with the explicitly identified external inputs F2–F4

1.1Given

Let $E$ be the compact exterior and $(M, γ)$ its cut along $F$ . Its two faces are copies of $F$ , and its vertical boundary is the annulus $\partial F \times I$ . It is balanced: both faces have Euler characteristic $1 - 2 g$ , no component is closed, and every boundary component meets the vertical annulus. A meridian intersects $F$ algebraically once, so $F$ is nonseparating and the cut is connected.

1.2F1

The signed cut comparison gives $SFH (M, γ; Z) ≅ \hat{H F K} (Y, K, [F], g; Z) ≅ Z$ . The grading uses the supplied class $[F]$ and genus of this actual surface; no genus-detection theorem for a different ambient manifold is substituted.

2.1Givenstep 1.1

The irreducibility assumption applies to $E$ : pushing away the deleted-knot end identifies every compact sphere question with that in $Y ∖ K$ . A sphere in the interior of $M$ is therefore a sphere disjoint from $F$ in $E$ and bounds a ball in $E$ . Since $F$ is connected and disjoint from this sphere, if it entered the ball it would lie wholly in its interior, contradicting $\partial F \subset \partial E$ . The ball is consequently disjoint from $F$ ; shrink the cut collar if necessary to obtain the same ball in $M$ . Thus $M$ is irreducible.

3.1F2step 1.1step 2.1step 1.2

If $(M, γ)$ were not taut, its integral sutured Floer homology would vanish by the external vanishing input, contradicting step 1.2. Therefore it is taut. This is the contrapositive of F2, not a reversal of taut hierarchy nonvanishing.

4.1F3F4step 1.2step 3.1∎

If the taut balanced $(M, γ)$ were not a product, F3 would give $Z^{2} ↪ SFH (M, γ; Z)$ . Such an injection into $Z$ is impossible: images $a, b$ of its two basis vectors satisfy the nontrivial integer relation $b a - a b = 0$ when either is nonzero, and an image zero itself contradicts injectivity. The two disjoint outer summands in the source argument therefore contradict step 1.2. Hence the actual $F$ -cut exterior is a product sutured manifold. This includes $g = 0$ and uses the corrected characteristic-pair qualification F4.

Depends on

Used by

Rank-one top knot Floer homology detects fibering Theorem

Dependency tree · two levels

17 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

Yi Ni, Knot Floer homology detects fibred knots, arXiv:math/0607156v4, §§3–7 (standard reference, not scraped)
Andras Juhasz, Floer homology and surface decompositions, arXiv:math/0609779v3, §§8–9 (standard reference, not scraped)
David Gabai, Foliations and the topology of 3-manifolds III, Theorem 3.13 and Theorem 8.9 (standard reference, not scraped)
András Juhász, Holomorphic discs and sutured manifolds, Algebraic & Geometric Topology 6 (2006), 1429–1457 (standard reference, not scraped)
András Juhász, Floer homology and surface decompositions, Geometry & Topology 12 (2008), 299–350 (standard reference, not scraped)
Yi Ni, Corrigendum to Knot Floer homology detects fibred knots, Inventiones Mathematicae 177 (2009), 235–238 (standard reference, not scraped)