On Expansion and Topological Overlap

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 $X\rightarrow \mathbf{R}^d$ 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 $M$.