Calculus of Variations and Geometric Measure Theory

E. Bruè - A. Naber - D. Semola

Fundamental Groups and the Milnor Conjecture

created by semola on 28 Mar 2023



Inserted: 28 mar 2023
Last Updated: 28 mar 2023

Pages: 56
Year: 2023

ArXiv: 2303.15347 PDF


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 ${\rm Ric}\geq 0$ 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 ${\rm Ric}>0$.