Kirby Problem 4.27

Definitions and mathematical statement

Let $B^3$ be the standard closed smooth $3$-ball. Consider smooth embeddings into the standard $S^4$ with images $B_0,B_1$ and a common boundary $S=\partial B_0=\partial B_1$, a smoothly embedded $2$-sphere. The interiors of the two balls are not required to be disjoint.

A topological ambient isotopy relative to $S$ is a continuous family of homeomorphisms $H_t:S^4\to S^4$, $0\le t\le1$, with $H_0$ the identity, $H_t|_S$ the identity for every $t$, and $H_1(B_0)=B_1$. A smooth ambient isotopy has the analogous properties with $(x,t) \mapsto H_t(x)$ smooth and each $H_t$ a diffeomorphism.

Do there exist $B_0,B_1$ which admit such a topological isotopy but admit no such smooth isotopy? This is the requested exotic pair of embedded $3$-balls relative to their common boundary. The distinction concerns their placements in $S^4$; each individual ball already has the standard smooth ball structure.

The boundary sphere is fixed pointwise throughout both sorts of isotopy. Merely finding two nonisotopic balls would not answer the question unless they can also be shown topologically isotopic under this boundary restriction. No fixed parameterizations of the interiors are required, and no prescribed geometry of the boundary sphere is assumed. The entire smooth-isotopy class, not just a specific attempted isotopy, must be excluded.

Short English statement

Can two smooth three-balls with the same boundary in the four-sphere be topologically but not smoothly ambient isotopic while fixing that boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2903 (KP-4.27), 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 4.27. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.