| Summary: | 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).
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