When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.35
URN: urn:nbn:de:0030-drops-59270
URL: https://drops.dagstuhl.de/opus/volltexte/2016/5927/
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### On Expansion and Topological Overlap

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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 -> R^d there exists a point p in R^d whose preimage intersects a positive fraction mu > 0 of the d-cells of X. More generally, the conclusion holds if 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.

### BibTeX - Entry

@InProceedings{dotterrer_et_al:LIPIcs:2016:5927,
author =	{Dominic Dotterrer and Tali Kaufman and Uli Wagner},
title =	{{On Expansion and Topological Overlap}},
booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
pages =	{35:1--35:10},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-009-5},
ISSN =	{1868-8969},
year =	{2016},
volume =	{51},
editor =	{S{\'a}ndor Fekete and Anna Lubiw},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},