  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 $M$.
