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Session S23 - Group actions in Differential Geometry

Friday, July 23, 20:20 ~ 20:50 UTC-3

Upper bound on the revised first Betti number and torus stability for RCD spaces

Raquel Perales

IMATE UNAM, Oaxaca , Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Gromov and Gallot showed in the past century that for a fixed dimension n there exists a positive number $\varepsilon(n)$ so that any $n$-dimensional riemannian manifold satisfying $Ric_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)$ has first Betti number smaller than or equal to $n$. Furthermore, by Cheeger-Colding if the first Betti number equals $n$ then $M$ is bi-Hölder homeomorphic to a flat torus. This part is the corresponding stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension has to be a torus. The proof of Gromov and Cheeger-Colding results rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of $M$ and then study the geometry of the covering. In this talk we will generalize these results to the case of $RCD(K,N)$ spaces, which is the synthetic notion of a riemannian manifold satisfying $Ric \geq K$ and $dim \leq N$. This class of spaces include ricci limit spaces and Alexandrov spaces.

Joint work with I. Mondello and A. Mondino.

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