Kirby Problem 1.58

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.

The zero-trace $X_0(K)$ of a knot $K\subset S^3$ is $B^4$ with a 2-handle attached along $K$ with framing zero relative to its Seifert longitude. Its boundary $S^3_0(K)$ is zero-surgery. A meridian is the loop linking $K$ once, regarded also as a loop in zero-surgery. Topological concordance means a locally flat embedded annulus in $S^3\times[0,1]$; smooth concordance requires a smooth annulus. A topological integral homology cobordism between closed 3-manifolds is a compact topological 4-manifold for which both boundary inclusions induce integral homology isomorphisms.

Statement recorded in the supplied source.

For knots $K,K'\subset S^3$, determine separately whether topological concordance follows from each of:

1. $X_0(K)\cong X_0(K')$ by a homeomorphism;
2. $S^3_0(K)\cong S^3_0(K')$ by a homeomorphism;
3. $S^3_0(K)$ and $S^3_0(K')$ being topologically integral-homology cobordant with their meridian classes identified in $H_1$.

Additionally, if a homeomorphism $S^3_0(K)\to S^3_0(K')$ carries the meridian of $K$ to a loop freely homotopic to the meridian of $K'$, must the knots be smoothly concordant? [S2]

Short English statement

Which of the zero-trace, zero-surgery, and meridian-preserving homology-cobordism conditions force topological concordance? Does preservation of the meridian free-homotopy class under a surgery homeomorphism force smooth concordance?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2717 (KP-1.58). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 1.58 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.