# An upper bound on the revised first Betti number and a torus stability result for RCD spaces

created by mondino on 13 Apr 2021
modified on 14 Apr 2021

[BibTeX]

Submitted Paper

Inserted: 13 apr 2021
Last Updated: 14 apr 2021

Year: 2021

Abstract:

We prove an upper bound on the rank of the abelianised revised fundamental group (called revised first Betti number'') of a compact $\mathsf{RCD}^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating that the space is $\lfloor N \rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $\lfloor N \rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-H\"older homeomorphic to a flat torus. This second result extends to the class of non-smooth $\mathsf{RCD}^{*}(-\delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).