Kirby Problem 4.13

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $B^4$ be the standard closed four-ball and let $K\subset S^3=\partial B^4$ be a smooth knot. Its zero trace $X_K$ is obtained by attaching a two-handle $D^2\times D^2$ to $B^4$ along $K$ with framing zero relative to the Seifert longitude. A smooth $R^4$-homeomorph is a smooth manifold $R$ homeomorphic to $\mathbb R^4$, allowing the standard smooth structure. It is large if it contains a compact subset admitting no smooth embedding into standard $\mathbb R^4$. Define \[ \mathcal R_K=\{[R]_{\mathrm{diff}}:R\text{ is an }R^4\text{-homeomorph and }X_K\text{ smoothly embeds in }R\}. \] The notation $[R]_{\mathrm{diff}}$ denotes a diffeomorphism class. A knot is smoothly slice if it bounds a smoothly embedded proper disk in $B^4$.

Statement recorded in the supplied source.

Answer both questions:

1. (a) For every large $R^4$-homeomorph $R$, does there exist a knot $K$ that is not smoothly slice such that $[R]_{\mathrm{diff}}\in\mathcal R_K$?
2. (b) Do there exist knots $K_1,K_2,\ldots$ such that $\mathcal R_{K_i}\ne\mathcal R_{K_j}$ for every $i\ne j$?

The second assertion requires pairwise different sets of smooth homeomorphs, rather than merely different knots or different embedded traces. [S2]

Short English statement

Does every large smooth manifold homeomorphic to Euclidean four-space contain the zero trace of a knot that is not smoothly slice, and can infinitely many different trace-embedding sets be distinguished?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2889 (KP-4.13). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.13 and Remarks, PDF page 200. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.