Kirby Problem 1.55

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.

Write $\nu(A)$ for an open tubular neighborhood of a submanifold $A$. A fibered knot has exterior a bundle over $S^1$ with compact oriented fiber $F$ and boundary-fixing monodromy $h:F\to F$. Capping $\partial F$ by a disk and extending $h$ by the identity gives $\widehat h:\widehat F\to\widehat F$. A compression body $H$ is obtained from $\partial_-H\times[0,1]$ by adding 1-handles on the upper side; a handlebody is the case $\partial_-H=\varnothing$. Its upper boundary is $\partial_+H$.

Use the source convention for ribbon concordance: $C\subset S^3\times[0,1]$ is an embedded annulus with end knots $K_i$ at height $i$, and projection to $[0,1]$ has no local maxima after isotopy. The concordance exterior is $(S^3\times[0,1])\setminus\nu(C)$. A homotopy-ribbon disk $D\subset V$ satisfies surjectivity of $\pi_1(\partial V\setminus\nu(\partial D))\to\pi_1(V\setminus\nu(D))$. An integer homology 3-sphere has the same integral homology as $S^3$.

Statement recorded in the supplied source.

For a ribbon concordance $C$ from a fibered $K_1$ to a fibered $K_0$, ask both:

1. Does $\widehat h_1$ extend to a diffeomorphism of a compression body $H$ with $\partial_+H=\widehat F_1$, $\partial_-H=\widehat F_0$, and lower-boundary restriction $\widehat h_0$?
2. Does the exterior of $C$ fiber over $S^1$ with compression-body fibers, inducing the given knot-exterior fibrations at its ends?

Separately, if a fibered knot in an integer homology 3-sphere bounds a homotopy-ribbon disk $D$ in a contractible smooth 4-manifold $V$, must $V\setminus\nu(D)$ fiber over $S^1$ with handlebody fibers? [S2]

Short English statement

Do ribbon concordances between fibered knots extend their capped monodromies over compression bodies and themselves have such fiberings? Must a homotopy-ribbon disk exterior in a contractible 4-manifold fiber by handlebodies?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2714 (KP-1.55). 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 1.55 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.