On expansion and topological overlap |
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Authors: | Dominic Dotterrer Tali Kaufman Uli Wagner |
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Institution: | 1.Department of Mathematics,University of Chicago,Chicago,USA;2.Department of Computer Science,Bar-Ilan University,Ramat Gan,Israel;3.IST Austria,Klosterneuburg,Austria |
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Abstract: | 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 \(X\rightarrow \mathbb {R}^d\) there exists a point \(p\in \mathbb {R}^d\) that is contained in the images of a positive fraction \(\mu >0\) of the d-cells of X. More generally, the conclusion holds if \(\mathbb {R}^d\) is replaced by any d-dimensional piecewise-linear manifold M, with a constant \(\mu \) that depends only on d and on the expansion properties of X, but not on M. |
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