Title: The distortion of the Reeb quotient Map
Speaker: Osman Berat Okutan (Ohio State University)
Abstract: Given a metric space $X$ and a function $f:X \rightarrow \mathbb{R}$, the Reeb construction gives metric a space $Xf$ together with a quotient map $X \rightarrow Xf$. Under suitable conditions $Xf$ becomes a metric graph and can therefore be used as a graph approximation to $X$. The Gromov-Hausdorff distance from $Xf$ to $X$ is bounded by the half of the metric distortion of the quotient map. In this paper we consider the case where $X$ is a compact Riemannian manifold and $f$ is an excellent Morse function. In this case we provide bounds on the distortion of the quotient map which involve the first Betti number of the original space and a novel invariant which we call thickness.
Seminar URL: https://tgda.osu.edu/