Kirby Problem 1.61

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.

An integer homology 3-sphere $Y$ satisfies $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$. Its Freedman filling $\Delta$ is a compact contractible topological 4-manifold with boundary $Y$. An integral homology cobordism $W$ between $Y$ and $S^3$ has boundary $Y\sqcup(-S^3)$ and homology-isomorphism boundary inclusions. Knot or link concordance in $W$ means disjoint locally flat embedded annuli with the prescribed boundary components. A homology 4-ball has the integral homology of $B^4$. A semi-free action of $\mathbb Z/n\mathbb Z$ has stabilizer either trivial or the entire group. Arbitrary topological knot embeddings and disks may be wild; no local-linearity assumption is imposed on the actions below.

Statement recorded in the supplied source.

Resolve all of the following topological questions.

1. Is each locally flat knot in every integer homology 3-sphere concordant to a knot in $S^3$ in an integral homology cobordism? Can the cobordism also be simply connected?
2. Does every topologically embedded knot in $Y$ bound an embedded disk in its Freedman filling $\Delta$, without requiring local flatness? Does such a disk at least exist in some homology 4-ball bounded by $Y$?
3. Can a link in $Y$ have every component individually concordant to a knot in $S^3$, yet fail to be concordant as a link to any link in $S^3$?
4. If a semi-free action of $\mathbb Z/n\mathbb Z$ on $Y$ fixes precisely a knot $K$, does it extend to a semi-free action on $\Delta$?

[S2]

Short English statement

Ask whether knots and links in homology spheres admit the stated topological concordances or disk fillings, including wild disks, and whether semi-free cyclic actions extend over the contractible filling.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2720 (KP-1.61). 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 1.61 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.