Kirby Problem 4.11

Definitions and mathematical statement

Let $M$ be a smooth $4$-manifold and $f:S^2\hookrightarrow M$ a smooth embedding with trivial normal bundle. Thus a tubular neighborhood $\nu f$ is diffeomorphic to $S^2\times D^2$. The Gluck-twisted manifold $M_f$ is formed by cutting out its interior and regluing by the boundary self-map $\tau(x,e^{i\theta})=(R_\theta x,e^{i\theta})$, where $R_\theta$ rotates $S^2$ once as $\theta$ traverses the circle. Compose $\tau$ with the original attaching map. Any existing boundary of $M$ remains outside this interior operation.

Part (a) asks for an orientable $M$ and such an $f$ for which $M_f$ and $M$ are homeomorphic but not diffeomorphic. Part (b) asks for an orientable $M$ and smooth embeddings $f,g:S^2\hookrightarrow M$, both with trivial normal bundle, which are homotopic as maps, but for which $M_f$ and $M_g$ are homeomorphic and not diffeomorphic.

A homotopy in (b) is a continuous map $S^2\times[0,1]\to M$ connecting the embeddings; its intermediate maps need not be embeddings. Thus homotopy is weaker than isotopy or concordance. In both parts, the smooth distinction is between entire resulting manifolds, with arbitrary diffeomorphisms allowed, rather than between marked surgery descriptions. No closedness or simple-connectivity assumption is added to the source's orientable ambient-manifold question. Merely changing homeomorphism type by a Gluck twist would not answer either part.

Short English statement

Can Gluck twisting change the smooth type while preserving the homeomorphism type in an orientable four-manifold, including twists along homotopic spheres?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2887 (KP-4.11), 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 4.11. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.