On Expansion and Topological Overlap

Abstract

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. 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 $X\to {\mathbf{R}}^d$ there exists a point $p\in {\mathbf{R}}^d$ that is contained in the images of a positive fraction $\mu >0$ of the $d$-cells of $X$. More generally, the conclusion holds if ${\mathbf{R}}^d$ is replaced by any $d$-dimensional piecewise-linear (PL) manifold $M$, with a constant $\mu$ that depends only on $d$ and on the expansion properties of $X$, but not on $M$.

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