LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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

Statement

Assume AC. Let $K \subset S^{3}$ be a smooth oriented knot and let $D \subset B^{4}$ be a smooth properly embedded disk with boundary $K$ . Let $W_{D}$ be the connected double cover of $B^{4}$ branched along $D$ . Then $W_{D}$ is a rational homology four-ball and its oriented boundary is the double branched cover $Σ_{2} (K)$ formed using the positive-meridian convention. This assertion concerns smooth disks only and makes no claim for locally flat topological disks.

Statement references

The positive-meridian convention for the cyclic branched cover

The smooth slice-disk cover homology result

Facts & Assumptions

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

[A1]

AC is assumed by the published smooth-disk cover theorem (The Axiom of Choice).

[F1]

The double cyclic branched cover $Σ_{2} (K)$ is obtained from the connected exterior cover whose monodromy sends a positive meridian to the generator of $Z / 2$ , by filling along the closed lift of the squared meridian. Its oriented branched map has local form $(s, z) \mapsto (s, z^{2})$ near $K$ (Cyclic branched cover of a knot).

[F2]

Under AC, the connected double cover of $B^{4}$ branched over a smooth properly embedded disk is a compact connected oriented smooth rational homology four-ball, and its boundary is the double cover of $S^{3}$ branched over the disk's boundary. The supplier constructs it using the orientation-preserving normal model $(x, z) \mapsto (x, z^{2})$ , with a product collar at the boundary (The double cover branched over a slice disk is a rational homology ball).

Proof

technique · apply the smooth-disk cover result and identify the boundary cover

Given: The data above.

1.1A1F2given

By [F2], the connected branched double cover $W_{D}$ has the rational homology of a point; smoothness and properness of the disk are exactly the hypotheses of that supplier.

2.1F1F2givenconstruct∎

By [F2], the restriction of the covering map to $\partial B^{4} = S^{3}$ is the double cover branched along $\partial D = K$ . Its exterior cover is connected because the local model $z \mapsto z^{2}$ gives nontrivial monodromy on a positive meridian. Thus its exterior cover and meridional filling are exactly those defining $Σ_{2} (K)$ in [F1]. The orientation pulled back off the branch locus induces the same boundary orientation, since the product collar in [F2] preserves the outward normal. Therefore $\partial W_{D} = Σ_{2} (K)$ as oriented manifolds. The supplier was used only for smooth disks, so no topological-disk conclusion follows.

Depends on

Used by

Double-cover correction terms obstruct smooth sliceness Corollary

Dependency tree · two levels

31 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)
Dale Rolfsen, Knots and Links (standard reference, not scraped)