Kirby Problem 1.60

Definitions and mathematical statement

For $R=\mathbb Z$ or $R=\mathbb Z/2$, an $R$-homology four-ball is a compact connected oriented smooth $4$-manifold $W$ with $H_*(W;R)\cong H_*(B^4;R)$. In this problem require additionally an orientation-preserving identification $\partial W\cong S^3$. A knot $K\subset S^3$ is smoothly slice in $W$ if, under this identification, it bounds a properly embedded smooth disk $D\subset W$. Its smooth four-genus $g_4(K)$ is the minimum genus of smooth connected oriented bounding surfaces in the standard $B^4$.

Does there exist a knot $K$ with $g_4(K)>0$ that is smoothly slice in an integer homology four-ball? Ask the same existence question for a $\mathbb Z/2$-homology four-ball. The manifold $W$ and disk may depend on the example and need not be simply connected. Requiring $\partial W=S^3$ ensures the same ordinary knot is being compared in both settings. A disk in a rational homology ball alone does not meet either stated homology condition; integral and mod-two acyclicity must be checked with their specified coefficient rings.

Short English statement

Can a knot fail to be smoothly slice in the standard ball yet bound a smooth disk in a ball with the same integral or mod-two homology?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2719 (KP-1.60), 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 1.60. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.