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