$\mathrm{RCD}^{*}(K,N)$ spaces are semi-locally simply connected

TL;DR

This paper proves that $ ext{RCD}^*(K,N)$ spaces are semi-locally simply connected, ensuring their universal covers are simply connected, thus extending previous results on Ricci limit spaces to a broader class of metric measure spaces.

Contribution

The paper establishes semi-local simple connectivity for $ ext{RCD}^*(K,N)$ spaces, generalizing earlier results from Ricci limit spaces to this wider setting.

Findings

Any $ ext{RCD}^*(K,N)$ space is semi-locally simply connected.

Universal cover of an $ ext{RCD}^*(K,N)$ space is simply connected.

Loops in small balls are contractible within larger balls.

Abstract

It was shown by Mondino-Wei that any $\mathrm{RCD}^{*}(K,N)$ space $(X,d,\mathfrak{m})$ has a universal cover. We prove that for any point $x \in X$ and $R>0$, there exists $r<R$ such that any loop in $B_r(x)$ is contractible in $B_R(x)$; in particular, $X$ is semi-locally simply connected and the universal cover of $X$ is simply connected. This generalizes the author's earlier work that any Ricci limit space is semi-locally simply connected.