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 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.

Joint work with I. Mondello and A. Mondino.