Fundamental groups and the Milnor conjecture

Abstract

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 M7 with Ric ≥ 0 such that π1(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 π0Diff(S3 × S3) and its relationship to Ricci curvature. In particular, a key point will be to show that the action of π0Diff(S3 × S3) on the standard metric gS3×S3 lives in a path connected component of the space of metrics with Ric > 0.