Fundamental Groups and the Milnor Conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929672474853376 |
|---|---|
| author | Bruè, Elia Naber, Aaron Semola, Daniele |
| author_facet | Bruè, Elia Naber, Aaron Semola, Daniele |
| contents | 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 $π_1(M)=\mathbb{Q}/\mathbb{Z}$ 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 $π_0\text{Diff}(S^3\times S^3)$ and its relationship to Ricci curvature. In particular, a key point will be to show that the action of $π_0\text{Diff}(S^3\times S^3)$ on the standard metric $g_{S^3\times S^3}$ lives in a path connected component of the space of metrics with ${\rm Ric}>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15347 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fundamental Groups and the Milnor Conjecture Bruè, Elia Naber, Aaron Semola, Daniele Differential Geometry 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 $π_1(M)=\mathbb{Q}/\mathbb{Z}$ 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 $π_0\text{Diff}(S^3\times S^3)$ and its relationship to Ricci curvature. In particular, a key point will be to show that the action of $π_0\text{Diff}(S^3\times S^3)$ on the standard metric $g_{S^3\times S^3}$ lives in a path connected component of the space of metrics with ${\rm Ric}>0$. |
| title | Fundamental Groups and the Milnor Conjecture |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2303.15347 |