Some topological results of Ricci limit spaces

TL;DR

This paper investigates the topology of Ricci limit spaces, establishing conditions under which these spaces are semi-locally simply connected and describing their universal covers, advancing understanding of their topological structure.

Contribution

It proves semi-local simple connectivity of Ricci limit spaces under bounded covering geometry and describes their universal covers when diameters are bounded.

Findings

Ricci limit spaces are semi-locally simply connected under certain conditions

A slice theorem for isometric pseudo-group actions is established

Universal covers of Ricci limit spaces are characterized with diameter bounds

Abstract

We study the topology of a Ricci limit space $(X,p)$, which is the Gromov-Hausdorff limit of a sequence of complete $n$-manifolds $(M_i, p_i)$ with $\mathrm{Ric}\ge -(n-1)$. Our first result shows that, if $M_i$ has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then $X$ is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of $X$ if $M_i$ has a uniform diameter bound.