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A note on the topological stability theorem from RCD spaces to Riemannian manifolds

Inspired by a recent work of Wang-Zhao, in this note we prove that for a fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an $\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff close to $M^n$, then there exists a regular homeomorphism $F$ from $X$ to $M^n$ such that $F$ is Lipschitz continuous and that $F^{-1}$ is Hölder continuous, where the Lipschitz constant of $F$, the Hölder exponent and the Hölder constant of $F^{-1}$ can be chosen arbitrary close to $1$. This is sharp in the sense that in general such a map cannot be improved to being bi-Lipschitz. Moreover if $X$ is smooth, then such a homeomorphism can be chosen as a diffeomorphism. It is worth mentioning that the Lipschitz-Hölder continuity of $F$ improves the intrinsic Reifenberg theorem for closed manifolds with Ricci curvature bounded below established by Cheeger-Colding. The Nash embedding theorem plays a key role in the proof.