Kirby Problem 4.23

Definitions and mathematical statement

Give $S^4\subset\mathbb R^5$ and $S^3\subset\mathbb R^4$ their standard smooth structures. Let $i:S^3\hookrightarrow S^4$ be a smooth embedding, and put $\Sigma=i(S^3)$. The complement has two components; denote their closures by $W_0$ and $W_1$. Smoothness of the embedded hypersurface gives each $W_j$ the structure of a compact smooth $4$-manifold with boundary $\Sigma$.

Must both $W_0$ and $W_1$ be diffeomorphic to the standard closed ball $B^4=\{x\in\mathbb R^4:\|x\|\le1\}$? This is the smooth four-dimensional Schoenflies question. The quantifier ranges over all smooth embeddings $i$, without any bound on a handle decomposition or on the critical points of a height function on $\Sigma$.

The requested diffeomorphisms are of compact manifolds with boundary, so they must extend smoothly up to the boundary. No particular map from $\Sigma$ to the standard $S^3$ is prescribed. Proving standardness only of the interiors, or obtaining a homeomorphism of the closed complements, would be a weaker conclusion.

Both sides are part of the assertion. A counterexample would be a smoothly embedded $3$-sphere for which at least one closed complementary piece is not a smooth ball. The embedding has codimension one; this question is different from asking whether every smoothly embedded $2$-sphere in $S^4$ is unknotted.

Short English statement

Are both closed complementary regions of every smoothly embedded three-sphere in the four-sphere standard smooth four-balls?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2899 (KP-4.23), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.23. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.