Kirby Problem 5.6

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.

For $n\geq2$, a PL branched covering $f:M\to S^n$ is a finite-to-one open surjective PL map that restricts to an ordinary covering away from its branch-value set $B\subset S^n$ of codimension two. Its degree is the number of sheets over any point of $S^n\setminus B$. An embedded branch locus is an embedded codimension-two submanifold; local flatness is an additional condition and is not imposed by that wording. A simple cover has meridian monodromy a transposition in the permutation group of the sheets. A meridian is a small loop linking the regular part of the branch locus once. Orientability of $B$ concerns its manifold components.

Statement recorded in the supplied source.

For every $n\geq2$ and every closed connected orientable PL $n$-manifold $M$:

1. (a) Does $M$ admit a degree-$n$ branched covering of $S^n$?
2. (b) Can such a degree-$n$ cover have an embedded codimension-two branch locus?
3. (c) In the embedded-locus case, can the cover additionally be chosen simple?
4. (d) In that same case, characterize when the branch locus can be chosen orientable.

These requirements retain degree $n$; a cover of larger degree does not settle them. [S2]

Short English statement

Can every closed orientable PL n-manifold cover its sphere with n sheets, and which such covers can have embedded, simple, or orientable branching?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3013 (KP-5.6). 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 5.6, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.