Kirby Problem 5.9

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.

Fix $n\geq p\geq1$. A smooth map germ $(\mathbb R^n,0)\to(\mathbb R^p,0)$ is singular when its derivative at the distinguished point has rank less than $p$. Germs are right-left equivalent if local source and target diffeomorphisms conjugate one to the other. Let $S$ be any specified set of such singular-germ equivalence classes. A smooth map $f:M\to\mathbb R^p$ is an $S$-map if its germ at every singular point represents a member of $S$; regular points impose no extra restriction.

Statement recorded in the supplied source.

For every choice of $n,p,S$ and smooth $n$-manifolds $M_1,M_2$ that are homeomorphic, does the existence of an $S$-map $M_1\to\mathbb R^p$ imply the existence of an $S$-map $M_2\to\mathbb R^p$? Equivalently, is the existence of such maps independent of the smooth structure on a fixed topological manifold? [S2]

Short English statement

Does changing the smooth structure of a manifold preserve the existence of a map into Euclidean space having only a prescribed set of singularities?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3016 (KP-5.9). 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 5.9, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.