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.[2][3]
In two dimensions has finitely generated fundamental group as a consequence that if for noncompact , then it is flat or diffeomorphic to , by work of Cohn-Vossen from 1935.[4][5]
In three dimensions the conjecture holds due to a noncompact with being diffeomorphic to or having its universal cover isometrically split. The diffeomorphic part is due to Schoen-Yau (1982)[6][5] while the other part is by Liu (2013).[7][5] Another proof of the full statement has been given by Pan (2020).[8][5]
In 2023 Bruè et al. disproved in two preprints the conjecture for six[9] or more[5] dimensions by constructing counterexamples that they described as "smooth fractal snowflakes". The status of the conjecture for four or five dimensions remains open.[3]