Kirby Problem 4.32

Definitions and mathematical statement

Let $\Sigma$ be a closed connected surface, with no orientability assumption. A topological embedding $f:\Sigma\to S^4$ is locally flat if near each point of its image the pair $(S^4,f(\Sigma))$ is homeomorphic to $(\mathbb R^4,\mathbb R^2\times\{0\})$ near the origin. This excludes wild local behavior but imposes no differentiability on $f$.

Call the embedding smoothable by ambient isotopy if there is a continuous path of homeomorphisms $H_t:S^4\to S^4$, beginning at the identity, such that $H_1(f(\Sigma))$ is the image of a smooth embedding of the surface into the standard smooth $S^4$. A smooth structure on the abstract surface may be used when describing this final smooth embedding.

Does there exist a locally flat embedding which is not smoothable in this sense? In particular, can such an example have $\Sigma=S^2$? The negative requirement rules out every topological ambient isotopy to every smooth representative, not just the possibility of smoothing the original parameterization in place.

The ambient smooth structure stays the standard one. Local flatness is required everywhere; a wild embedding would not give the intended example. The general question allows all closed surface types, whereas the sphere case is a separate, stronger existence request. Obstructions available for a surface in some other smooth $4$-manifold would need to be realized in $S^4$ to answer this problem.

Short English statement

Can a locally flat closed surface in the standard four-sphere fail to be topologically ambient isotopic to every smooth surface, even when it is a sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2908 (KP-4.32), 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 4.32. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.