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

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Double-cover correction terms obstruct smooth sliceness

Statement

Assume AC. Let $K \subset S^{3}$ be a smooth oriented knot admitting a smooth properly embedded disk $D \subset B^{4}$ (a smooth embedding of the closed disk with interior in $int B^{4}$ and boundary $K \subset S^{3}$ ). Let $W_{D}$ be its connected double cover branched along $D$ , so $H_{*} (W_{D}; Q) ≅ H_{*} (p o i n t; Q)$ , and let $Y = Σ_{2} (K)$ be the oriented double branched cover of $S^{3}$ using the positive-meridian convention. Then $H_{*} (Y; Q) ≅ H_{*} (S^{3}; Q)$ . Define the torsion linking pairing $λ_{Y}$ on $H_{1} (Y; Z)$ by $λ_{Y} ([C], [D]) = (F \cdot D) / n (m o d Z)$ when $\partial F = n C$ , with signed intersection in the oriented $Y$ . Under Poincare duality, the restriction image of ${Spin}^{c} (W_{D})$ in ${Spin}^{c} (Y)$ is an affine coset of a subgroup $M \subseteq H_{1} (Y; Z)$ with $M = M^{⊥}$ , where $M^{⊥} = {x : λ_{Y} (m, x) = 0 for every m \in M}$ ; such a subgroup is a metabolizer. It has $∣ M ∣^{2} = ∣ H_{1} (Y; Z) ∣$ . Every extending structure $t$ in this coset satisfies $d (Y, t) = 0$ . No conclusion is asserted for a merely locally flat topological disk (a topological embedding with local flat disk charts but no smoothness hypothesis) or for nonextending boundary structures.

Statement references

The smooth slice-disk double cover is a rational homology ball

The extension coset is a self-annihilating subgroup

Correction-term vanishing for extending structures

Facts & Assumptions

Given: AC, a smooth oriented knot $K \subset S^{3}$ , and a smooth proper slice disk $D \subset B^{4}$ .

[A1]

AC is assumed by the smooth slice-cover and extension-set suppliers (The Axiom of Choice).

[F1]

The connected cover branched over $D$ is a rational homology four-ball and its oriented boundary is the connected cyclic double branched cover $Y = Σ_{2} (K)$ (The smooth slice-disk double cover is a rational homology ball, Cyclic branched cover of a knot).

[F2]

Poincaré–Lefschetz duality and the cohomology long exact sequence of $(W_{D}, Y)$ compute the rational cohomology of $Y$ from that of $W_{D}$ . Universal coefficients and closed-manifold Poincaré duality identify it with the rational homology of $S^{3}$ (Poincaré–Lefschetz duality, Long exact sequence of a pair in singular cohomology, The universal coefficient theorem for cohomology over a PID, Poincaré duality for oriented topological manifolds).

[F3]

For a rational ball with connected rational-homology-sphere boundary, the extending Spin $^{c}$ structures form one affine coset of a self-annihilating subgroup for the chain linking pairing, and that subgroup has square-root order (Spin-c extensions over a rational ball form a metabolizer coset).

[F4]

A Spin $^{c}$ structure extending over a smooth rational homology four-ball has correction term zero (Correction terms vanish on extending Spin-c structures over rational homology balls).

Proof

technique · apply the slice-cover, extension-set, and vanishing results in sequence

Given: The data above.

1.1A1F1given

Apply [F1] to $D$ . It gives a connected oriented rational homology four-ball $W_{D}$ with connected boundary $Y$ .

2.1F2step 1.1algebra

Since $H_{*} (W_{D}; Q) ≅ H_{*} (p o i n t; Q)$ , universal coefficients give $H^{0} (W_{D}; Q) = Q$ and $H^{j} (W_{D}; Q) = 0$ for $j > 0$ . By [F2], $H^{j} (W_{D}, Y; Q)$ is zero except for $H^{4} (W_{D}, Y; Q) ≅ Q$ . The pair sequence and connectedness of $Y$ give $H^{1} (Y; Q) = H^{2} (Y; Q) = 0$ and $H^{3} (Y; Q) ≅ Q$ ; also $H^{0} (Y; Q) ≅ Q$ . Closed-manifold Poincaré duality now gives $H_{*} (Y; Q) ≅ H_{*} (S^{3}; Q)$ .

3.1F3step 1.1step 2.1

Apply [F3] to $(W_{D}, Y)$ . The restriction image is one affine coset of a self-annihilating subgroup $M$ for the chain torsion linking pairing, and $∣ M ∣^{2} = ∣ H_{1} (Y; Z) ∣$ . The hypotheses hold by steps 1.1 and 2.1.

4.1F1F4step 3.1∎

Every structure in the restriction image of step 3.1 extends over $W_{D}$ , so [F4] gives $d (Y, t) = 0$ for each such structure. The proof used a smooth disk in [F1]; it does not apply to a merely locally flat topological disk. It makes no statement about structures outside the restriction image.

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62 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

Ralph H. Fox and John W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots (standard reference, not scraped)
Paolo Aceto and Marco Golla, Dehn Surgeries and Rational Homology Balls (standard reference, not scraped)
Peter S. Ozsvath and Zoltan Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary (standard reference, not scraped)