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

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The CDR smooth four-ball complement under the smooth four-sphere hypothesis

Statement

Let B be a smooth 4-ball embedded in the standard smooth 4-sphere S⁴. The closure of S⁴\B is diffeomorphic to the standard 4-ball. This is the special smooth Schoenflies complement assertion used after a capped contractible 4-manifold has been identified with standard S⁴; it does not assert the smooth 4-dimensional Poincaré conjecture.

Statement references

CDR, proof of Proposition 3.2

Facts & Assumptions

Given: A smooth embedded standard four-ball $B$ in the standard smooth sphere $S^{4}$ .

[F1]

The complement of a smoothly embedded standard four-ball in the standard smooth four-sphere is standard, by shrinking the whole parametrized ball and extending its isotopy. A standard smooth ball embedded in the standard four-sphere has standard complement

Proof

1.1givenconstruct

Choose a diffeomorphism $b : B^{4} \to B$ , where the existence of this parametrization is part of saying that $B$ is a smooth four-ball. Compose it with the given embedding into $S^{4}$ to obtain a smooth embedding $f : B^{4} ↪ S^{4}$ . This supplies an embedded standard side, rather than merely a boundary three-sphere or a contractible complementary manifold.

2.1F1step 1.1∎

Apply [F1] to $f$ . It gives a diffeomorphism from $\overline{S^{4} ∖ f (B^{4})} = \overline{S^{4} ∖ B}$ to the standard $B^{4}$ , as claimed. No smooth Poincaré assertion is needed for this complement step; any earlier identification of a capped ambient manifold with standard $S^{4}$ is a separate hypothesis of the application.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Tim D. Cochran, Christopher W. Davis, and Arunima Ray, “Injectivity of satellite operators in knot concordance” (standard reference, not scraped)