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On Expansion and Topological Overlap.

Banff International Research Station for Mathematical Innovation and Discovery

We present a simple and fairly elementary proof of Gromov?s Topological Overlap Theorem. Let $X$ is a finite $d$-dimensional simplicial complex. Informally, the theorem states that if $X$ has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of $X$) then $X$ has the following topological overlap property: for every continuous map from $X$ to $d$-dimensional Euclidean space, there exists an image point p contained in the images of a positive fraction $\mu>0$ of the $d$-simplices of $X$. More generally, the conclusion holds if d is replaced by any $d$-dimensional piecewise-linear manifold $M$, with a constant $\mu$ that depends only on d and on the expansion properties of $X$, but not on $M$. Joint work with Dominic Dotterrer and Tali Kaufman.

Mathematics; Convex and discrete geometry; Algebraic topology; Combinatorics

video/mp4

Author affiliation: IST Austria

2017-06-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61879

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/