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SUMMARY:Raquel Perales (National Autonomous University of Mexico)
DTSTART:20210519T160000Z
DTEND:20210519T170000Z
DTSTAMP:20260513T213312Z
UID:VSGS/31
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VSGS/31/">Up
 per bound on the revised first Betti number and torus stability for RCD sp
 aces</a>\nby Raquel Perales (National Autonomous University of Mexico) as 
 part of Virtual seminar on geometry with symmetries\n\n\nAbstract\nGromov 
 and Gallot showed in the past century that for a fixed dimension n there e
 xists a positive number $\\varepsilon(n)$ so that any $n$-dimensional riem
 annian manifold satisfying $Ric_g \\textrm{diam}(M\,g)^2 \\geq -\\varepsil
 on(n)$ 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-H
 ölder homeomorphic to a flat torus.  This part is the corresponding stabi
 lity statement to the rigidity result proven by Bochner\, namely\, closed 
 riemannian manifolds with nonnegative Ricci curvature and first Betti numb
 er equal to their dimension has to be a torus. \n\nThe proof of Gromov and
  Cheeger-Colding results rely on finding an appropriate subgroup of the ab
 elianized fundamental group to pass to a nice covering space of $M$ and th
 en study the geometry of the covering.   In this talk we will generalize t
 hese results to the case of $RCD(K\,N)$ spaces\, which is the synthetic no
 tion of a riemannian manifold satisfying $Ric \\geq K$  and $dim \\leq N$.
  This class of spaces include ricci limit spaces and Alexandrov spaces. \n
 \n Joint work with I. Mondello and A. Mondino.\n
LOCATION:https://researchseminars.org/talk/VSGS/31/
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