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 $M^7$ with $\mathrm{Ric}\ge 0$ such that ${\pi }_1(M)=\mathbb{Q}/\mathbb{Z}$ 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 ${\pi }_0\text{Diff}(S^3\times S^3)$ and its relationship to Ricci curvature. In particular, a key point will be to show that the action of ${\pi }_0\text{Diff}(S^3\times S^3)$ on the standard metric $g_{S^3\times S^3}$ lives in a path connected component of the space of metrics with $\mathrm{Ric}>0$.

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Cited by in corpus (2)

• Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity
• Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups