Some topological results of Ricci limit spaces

Abstract

We study the topology of a Ricci limit space ( X , p ) (X,p) , which is the Gromov-Hausdorff limit of a sequence of complete n n -manifolds ( M i , p i ) (M_i, p_i) with R i c ≥ − ( n − 1 ) \mathrm {Ric}\ge -(n-1) . Our first result shows that, if M i M_i has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then X 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 X if M i M_i has a uniform diameter bound; this improves a result by Ennis and Wei [Differential Geom. Appl. 24 (2006), pp. 554-562].