Kirby Problem 4.38

Definitions and mathematical statement

Give $B^4\subset\mathbb R^4$ its standard smooth structure and let $\Delta\subset B^4$ be a properly smoothly embedded disk, with its boundary a knot in $S^3$. It is a ribbon disk if, after a suitable smooth isotopy, a radial Morse function on the disk has no interior local maxima. Its interior critical points may have index zero or one only. Equivalently the disk is built from disjoint unknotted minimum disks by attaching bands, with the ribbon embedding data.

The question concerns the complement $B^4\setminus\Delta$. One may instead use the exterior obtained by deleting the interior of a small tubular neighborhood of the disk and rounding boundary corners; this has the same homotopy type. A connected space is aspherical if \[ \pi_i(X,x)=0\qquad\text{for every integer }i\ge2, \] for one, hence every, basepoint. For spaces with the homotopy type of a CW complex this is equivalent to contractibility of the universal cover.

Is the complement of every ribbon disk aspherical? The boundary knot is arbitrary, subject to having the specified ribbon disk. No extra condition such as local indicability is imposed on the complement's fundamental group. The assertion concerns each ribbon disk, not merely the existence of one aspherical disk for a given knot. Contractibility of the disk itself does not imply asphericity of its complement; the latter depends on the embedding.

Short English statement

Is the complement of every smooth ribbon disk in the four-ball aspherical?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2914 (KP-4.38), 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 4.38. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.