Milnor's conjecture on fundamental groups

Milnor’s conjecture asks whether complete manifolds with nonnegative Ricci curvature must have finitely generated fundamental groups. The August 2026 preprint claims counterexamples in dimensions four and five, extending the earlier seven-dimensional construction.

Known results

• Bruè, Naber, and Semola (2023): for every subgroup $\Gamma\leq\mathbb{Q}/\mathbb{Z}$, there is a complete $7$-manifold with $\operatorname{Ric}\geq0$ and $\pi_1(M)=\Gamma$.
• The same work left finite generation open for dimensions $4$, $5$, and $6$.
• Conditional results prove finite generation under hypotheses at infinity, including Euclidean volume growth with a unique tangent cone.
• A 2025 result proves finite generation for open $4$-manifolds with nonnegative Ricci curvature whose universal covers have Euclidean volume growth.

August 2026 claimed counterexamples

Nan Wu and Zetian Yan claim that, for $m\in\{2,3\}$, there is a complete, orientable, one-ended manifold of dimension $m+2$ with $\operatorname{Ric}_g>0$ and $\pi_1(M)\cong C_{2^\infty}$. This would settle the remaining dimensional cases by disproving finite generation in dimensions $4$ and $5$. The version-1 preprint reports author use of ChatGPT/Codex for assistance, not for producing the theorem.

Current status (as of August 2026): The $7$-dimensional counterexample and several conditional results are established, while the claimed $4$- and $5$-dimensional counterexamples remain unverified.