Fundamental groups and the Milnor conjecture

• 成果类型: Article
• 署名作者: Brue, Elia; Naber, Aaron; Semola, Daniele
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2025.201.1.4
• 发表日期: 2025
• 页码: 225-289
• 关键词: nonnegative ricci curvature; complete manifolds; finite generation; tangent-cones; ELEMENTS; METRICS; LIMITS; loops

摘要

It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example M-7 with Ric >= 0 such that pi 1(M) = Q/Z is infinitely generated. There are several new points behind the result. The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake. The ability to build such a fractal structure will rely on a very twisted gluing mechanism. Thus the other new point is a careful analysis of the mapping class group pi 0Diff(S-3 x S-3) and its relationship to Ricci curvature. In particular, a key point will be to show that the action of pi 0Diff(S3 x S3) on the standard metric gS(3)xS(3) lives in a path connected component of the space of metrics with Ric > 0.