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→ℝ^d there exists a point p∈ℝ^d that is contained in the images of a positive fraction μ >0 of the d -cells of X . More generally, the conclusion holds if ℝ^d is replaced by any d -dimensional piecewise-linear manifold M , with a constant μ that depends only on d and on the expansion properties of X , but not on M .