**Raquel Perales, IMATE-UNAM Oaxaca**

**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
diam^{2} 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.

Joint work with I. Mondello and A. Mondino.