Ricci limit spaces are semi-locally simply connected (Q6633923)

Ricci limit spaces are semi-locally simply-connected. This result generalizes previous work of \textit{J. Pan} and \textit{J. Wang} [Trans. Am. Math. Soc. 375, No. 12, 8445--8464 (2022; Zbl 1506.53053)], \textit{J. Pan} and \textit{G. Wei} [J. Eur. Math. Soc. (JEMS) 24, No. 12, 4027--4062 (2022; Zbl 1508.53047)], \textit{C. Sormani} and \textit{G. Wei} [Trans. Am. Math. Soc. 353, No. 9, 3585--3602 (2001; Zbl 1005.53035); Trans. Am. Math. Soc. 356, No. 3, 1233--1270 (2004; Zbl 1046.53027)]. \N\NThe main result of this paper is slightly stronger: for any point \(p\) in a Ricci limit space and any \(\varepsilon > 0 \), there is \(\delta > 0 \) such that any loop in \(B_{\delta}(p)\) is nullhomotopic in \(B_{\varepsilon}(p)\). The proof of this result combines new ideas with techniques previously established in [Pan and Wei, loc. cit.] and a slice theorem for local group actions obtained in [Pan and Wang, loc. cit.].\N\NThe main technical tool required to prove the main result is Lemma 3.1. It is very general and avoids working on separate cases as in [Pan and Wei, loc. cit.]. It establishes that for a sequence of pointed \(n\)-dimensional Riemannian manifolds \((M_i,p_i)\) sharing a lower Ricci curvature bound and converging in the pointed Gromov-Hausdorff sense to a space \((X,p)\), there is a small \(r > 0\) for which any loop \(\gamma\) in \(B_r(p)\) can be approximated by loops \(\gamma _i\) in \(B_r(p_i)\) that are approximately nullhomotopic. In here, approximately nullhomotopic means homotopic to arbitrarily short loops \(\gamma _i '\) based at \(p_i\) in such a way that the images of the homotopies have controlled diameter.\N\NWith Lemma 3.1, one would like to ``push'' the homotopies between \(\gamma _i\) to \(\gamma _i '\) to a homotopy between \(\gamma \) and a constant loop. It had been pointed out that this cannot work [Pan and Wei, loc. cit.]. However, using these homotopies one can break down \(\gamma\) into many much smaller loops, to which one can again apply Lemma 3.1. Iterating this process infinitely many times, one can produce a nullhomotopy of \(\gamma\). This part of the proof is an adaptation of [Pan and Wei, loc. cit.], where this idea is used to tackle a particular case of the main problem.