Kirby Problem 4.10

Definitions and mathematical statement

A smooth homotopy $4$-ball is a compact smooth $4$-manifold $X$ homotopy equivalent to a point; here its boundary is additionally required to be diffeomorphic to $S^3$. Let $K:S^2\hookrightarrow \operatorname{int}(B^4)$ be a smooth embedding. Its tubular neighborhood is $S^2\times D^2$. The Gluck operation replaces this neighborhood using the boundary map \[ (x,e^{i\theta})\longmapsto(R_\theta x,e^{i\theta}), \] where $R_\theta$ makes one complete rotation about an axis in $S^2$. Equivalently this is the loop generating $\pi_1(SO(3))$. The original outer boundary $\partial B^4=S^3$ is untouched.

Part (a) asks whether every smooth homotopy $4$-ball with boundary $S^3$ is diffeomorphic to a manifold obtained by one such Gluck twist inside $B^4$. The embedding may depend on the ball, and may be knotted. The comparison is a diffeomorphism of manifolds with boundary; no preassigned boundary parameterization is required in the statement.

Part (b) asks whether every $X$ obtained by that construction satisfies $X\times[0,1]\cong B^5$ smoothly, after rounding the corners of the product. The question is about a $5$-dimensional smooth ball, not just a contractible product. It does not require the diffeomorphism to preserve the product coordinate. Even if the product is standard, that conclusion alone is weaker than asserting that $X$ itself is a standard smooth $4$-ball.

Short English statement

Are all smooth homotopy four-balls with sphere boundary single Gluck twists of the standard ball, and do all such twists become a standard five-ball after taking a product with an interval?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2886 (KP-4.10), 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 4.10. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.