In 1968 John Milnor conjectured[1] that the fundamental group of a complete manifold is finitely generated if its Ricci curvature stays nonnegative.
In an oversimplified interpretation, such a manifold has a finite number of "holes".
A version for almost-flat manifolds holds from work of Gromov.
has finitely generated fundamental group as a consequence that if
[4][5] In three dimensions the conjecture holds due to a noncompact
[8][5] In 2023 Bruè, Naber and Semola disproved in two preprints the conjecture for six[9] or more[5] dimensions by constructing counterexamples that they described as "smooth fractal snowflakes".