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

TL;DR

This paper establishes an upper bound on the revised first Betti number of compact RCD* spaces and proves a torus stability result when this bound is saturated, extending classical results to non-smooth spaces.

Contribution

It provides the first upper bound on the revised first Betti number for RCD* spaces and extends torus stability theorems to non-smooth metric measure spaces.

Findings

Upper bound on the revised first Betti number for RCD* spaces.

Torus stability when the bound is saturated, with spaces close to flat tori.

Bi-Hölder homeomorphism to flat tori when N is an integer.

Abstract

We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ 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 $RCD^{*}(-\delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).