Kirby Problem 5.15

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.

Let $(M,g)$ be a connected smooth Riemannian manifold without boundary. Completeness means every geodesic extends for all time, equivalently the associated metric is complete. Nonnegative Ricci curvature means $\operatorname{Ric}_g(v,v)\geq0$ for every tangent vector $v$. The fundamental group $\pi_1(M)$ is finitely generated if a finite set of its elements generates it as a group.

Statement recorded in the supplied source.

For every dimension $n$ and every complete $n$-dimensional Riemannian manifold $(M,g)$ with nonnegative Ricci curvature, is $\pi_1(M)$ finitely generated? Preserve the universal question, and in particular determine its remaining cases in dimensions four and five, as singled out in the source. The source records counterexamples in dimensions six and higher. [S2]

Short English statement

Does complete nonnegative Ricci curvature force a finitely generated fundamental group, particularly in the unresolved dimensions four and five?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3022 (KP-5.15). 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 5.15, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.