On Expansion and Topological Overlap
Dominic Dotterrer, Tali Kaufman, Uli Wagner

TL;DR
This paper provides an accessible proof of Gromov's Topological Overlap Theorem, linking high-dimensional expansion properties of complexes to the guaranteed overlap of images under continuous maps into Euclidean spaces or manifolds.
Contribution
It offers a detailed, accessible proof of Gromov's theorem, connecting expansion properties of complexes to topological overlap phenomena in Euclidean spaces and manifolds.
Findings
High-dimensional expansion implies topological overlap
Overlap point is contained in a positive fraction of cells
Results apply to complexes mapped into manifolds
Abstract
We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension . Informally, the theorem states that if has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of ) then has the following topological overlap property: for every continuous map there exists a point that is contained in the images of a positive fraction of the -cells of . More generally, the conclusion holds if is replaced by any -dimensional piecewise-linear (PL) manifold , with a constant that depends only on and on the expansion properties of , but not on .
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