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Knotted 3-balls in S^4

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arxiv 1912.09029 v3 pith:PDWPFIZ5 submitted 2019-12-19 math.GT math.AT

Ryan Budney , David Gabai This is my paper · ORCID
classification math.GTmath.AT
keywords ballsdiffeomorphismsintroduceisotopicmanifoldsmethodnon-separatingparameter
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The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy fixing U. Equivalently there are properly embedded non-separating 3-balls in S^1xB^3 not properly isotopic to 1xB^3. More generally there exist non-separating 3-spheres in S^1xS^3 not isotopic to 1xS^3 and non trivial elements of \pi_0 Diff_0(S^1xS^3). Along the way we introduce barbell diffeomorphisms, implantations and twistings to construct and modify diffeomorphisms homotopic to the identity. We also introduce a 2-parameter calculus of embeddings of the interval into 4-manifolds and introduce a framed cobordism method as well as a direct method for showing that certain 2-parameter families are homotopically non trivial and diffeomorphisms are isotopically nontrivial. Extensions to higher dimensional manifolds are obtained.

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Cited by 4 Pith papers

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    A handle-exchange construction produces new open book decompositions in all dimensions n≥3 and shows every trivial-monodromy open book stabilizes to a page made of trivial disk bundles over spheres.

  4. On the Dax invariants of $S^2$-bundles over surfaces

    math.GT 2026-07 accept novelty 5.5 of 10

    Relative Dax invariants classify dual-paired embeddings of Σ into the nontrivial S^{2}-bundle Σ ⋉ S^{2} up to isotopy and yield a surjective homomorphism MCG(Σ ⋉ S^{2}) → ℤ^∞.

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