Kirby Problem 4.122

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.

Work in the smooth category with compact orientable manifolds and smoothly embedded branch surfaces, following the original construction. A finite branched covering $p:W\to N$ with branch locus $F\subset N$ is an ordinary finite covering off $F$, with local topological model $(z_1,z_2)\mapsto(z_1^e,z_2)$ along the branch surface; degrees and branching indices may depend on $W$, and simplicity is not required. The branch locus is the image of the singular set. A properly embedded compact surface $F\subset B^4$ is ribbon if the radial function restricts to a Morse function with no local maximum in its interior. Call $F$ universal in $B^4$ when every compact orientable smooth four-manifold built from $B^4$ by attaching only one- and two-handles is a finite branched cover of $B^4$ with branch locus exactly $F$. A universal link $L\subset S^3$ is a fixed smooth link such that every closed orientable three-manifold is a finite cover of $S^3$ branched over exactly $L$. [S3]

Statement recorded in the supplied source.

Is there a single smoothly embedded surface $S\subset S^4$, allowed to be disconnected, such that every closed orientable smooth four-manifold $W$ admits a finite branched covering $W\to S^4$ with branch locus exactly $S$? In the associated questions, does a connected universal surface exist in $B^4$, and which universal links in $S^3$ occur as $\partial F$ for universal surfaces $F\subset B^4$? [S2][S3]

Short English statement

Can one fixed surface in the four-sphere serve as the branch locus for every closed orientable smooth four-manifold? Also seek connected universal surfaces in the four-ball and characterize which universal three-dimensional links bound universal surfaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2998 (KP-4.122). 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, AMS Mathematical Surveys and Monographs 295 (2026), author preliminary edition. Problem4.122 and all four remarks, pages291--292. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Riccardo Piergallini and Daniele Zuddas, A universal ribbon surface in $B^4$, arXiv:math/0308222v1 (2003); Proceedings of the London Mathematical Society90 (2005),763--782. Introduction, page2 definitions, Section4 Questions2 and4. https://arxiv.org/pdf/math/0308222v1.