Kirby Problem 4.9

Definitions and mathematical statement

A smooth $2$-knot is a smooth embedding $K:S^2\hookrightarrow S^4$. Its normal bundle is trivial, so a tubular neighborhood can be identified with $S^2\times D^2$. Write $E_K=S^4\setminus\operatorname{int}(\nu K)$, whose boundary is $S^2\times S^1$ after this identification. Let $R_\theta\in SO(3)$ be rotation through angle $\theta$ around a fixed axis, for $0\le\theta\le2\pi$. Define \[ \tau(x,e^{i\theta})=(R_\theta x,e^{i\theta}). \] The loop of rotations represents the nontrivial element of $\pi_1(SO(3))\cong\mathbb Z/2$.

The Gluck twist $X_K$ is obtained by removing $\nu K$ and attaching $S^2\times D^2$ again, composing its original boundary identification with $\tau$. Gluing the smooth boundary collars gives a closed smooth $4$-manifold. Is $X_K$ diffeomorphic to the standard $S^4$ for every smooth $2$-knot $K$?

Standardness here concerns the ambient manifold after surgery. It does not assert that $K$ itself is an unknotted sphere, nor that a resulting diffeomorphism respects the removed neighborhood or its core sphere. The twist is the specified rotating boundary map, not an arbitrary regluing of $S^2\times S^1$. The domain of the question includes all smoothly knotted spheres, without a ribbon, spinning, or symmetry assumption. A homeomorphism to $S^4$ is weaker than the requested smooth identification.

Short English statement

Does Gluck surgery on every smoothly embedded two-sphere in the standard four-sphere produce the standard smooth four-sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2885 (KP-4.9), 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 4.9. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.