Kirby Problem 4.30

Definitions and mathematical statement

For $g\ge0$, let $\Sigma_g$ be the closed connected oriented smooth surface of genus $g$. Consider smooth embeddings $e:\Sigma_g\hookrightarrow S^4$, where the ambient sphere has its standard smooth structure. Their images carry the orientations induced by the embeddings. Two such surfaces are topologically ambient isotopic if a path of homeomorphisms $H_t:S^4\to S^4$ starting at the identity sends one oriented image to the other at $t=1$. They are smoothly ambient isotopic if the same can be done by a smooth family of diffeomorphisms.

Does there exist a pair which is topologically but not smoothly ambient isotopic? If a pair exists, does there exist an infinite family of pairwise smoothly nonisotopic oriented surfaces, all in one topological ambient-isotopy class? Can the pair or family consist of $2$-spheres, that is, have genus zero? More generally, can the required phenomenon occur for each prescribed integer $g\ge0$?

A single topological isotopy class fixes the genus, so an infinite family must not obtain its distinction merely by changing surface topology. The requested ambient manifold is exactly $S^4$, and the surfaces must be orientable. Examples in other $4$-manifolds or using nonorientable surfaces do not meet those conditions. The equivalence here concerns oriented embedded images; it does not require a specified parameterization of the surface to be carried pointwise along the isotopy.

Short English statement

Are there orientable surfaces in the four-sphere that are topologically but not smoothly isotopic, including infinite families and every prescribed genus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2906 (KP-4.30), 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 4.30. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.