RCD*(K,N) spaces are semi-locally simply~connected

Abstract

Abstract It was shown in [A. Mondino and G. Wei, On the universal cover and the fundamental group of an RCD * ⁢ ( K , N ) {\rm RCD}^{*}(K,N) -space, J. reine angew. Math. 753 2019, 211–237] that any RCD * ⁢ ( K , N ) {\mathrm{RCD}^{*}(K,N)} space ( X , d , 𝔪 ) {(X,d,\mathfrak{m})} has a universal cover. We prove that for any point x ∈ X {x\in X} and R > 0 {R>0} , there exists r < R {r<R} such that any loop in B r ⁢ ( x ) {B_{r}(x)} is contractible in B R ⁢ ( x ) {B_{R}(x)} ; in particular, X is semi-locally simply connected and the universal cover of X is simply connected. This generalizes the earlier work in [J. Wang, Ricci limit spaces are semi-locally simply connected, preprint 2021] that any Ricci limit space is semi-locally simply connected.