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On Expansion and Topological Overlap. Wagner, Uli


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.

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