On Expansion and Topological Overlap

We present a detailed and easily accessible proof of Gromov's \emph{Topological Overlap Theorem}: Let $X$ is a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $\dim X=d$. Informally, the theorem states that if $X$ has sufficiently strong \emph{higher-dimensional expansion properties} (which generalize edge expansion of graphs and are terms of cellular cochains of $X$) then $X$ has the following \emph{topological overlap property}: for every continuous map $X \rightarrow R^d$ there exists a point $p\in R^d$ whose preimage meets a positive fraction $\mu>0$ of the $d$-cells of $X$. More generally, the conclusion holds if $R^d$ 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 $M$.