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

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