تقرير
Fundamental Groups and the Milnor Conjecture
| العنوان: | Fundamental Groups and the Milnor Conjecture |
|---|---|
| المؤلفون: | Bruè, Elia, Naber, Aaron, Semola, Daniele |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry |
| الوصف: | 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 ${\rm Ric}>0$. Comment: Minor edits |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2303.15347 |
| رقم الأكسشن: | edsarx.2303.15347 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |