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

Abstract

It was shown by Mondino-Wei that any ${\mathrm{RCD}}^{\ast }(K,N)$ 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.