Title: Upper bound on the revised first Betti number and torus
stability for RCD spaces
Abstract: It was shown by Gromov and Gallot that for a fixed
dimension n there exists a positive number ε(n) so that
any n-dimensional riemannian manifold M satisfying
diam2 Ric > - ε(n) must have first Betti
number smaller than or equal to n. Later on, Cheeger and Colding
showed that if the first Betti number equals n then M has to be
bi-Hölder homeomorphic to a flat torus.
In this talk we will generalize the previous results to the case of
RCD(K,N) spaces, which is the synthetic notion of
riemannian manifolds satisfying the inequalities Ric ≥ K and
dim ≤ N. This class of spaces include Ricci limit spaces
and Alexandrov spaces.