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 → ℝd there exists a point p ∈ ℝd whose preimage intersects a positive fraction μ > 0 of the d-cells of X. More generally, the conclusion holds if ℝd is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant μ that depends only on d and on the expansion properties of X, but not on M.
Original language | English |
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Title of host publication | 32nd International Symposium on Computational Geometry, SoCG 2016 |
Editors | Sandor Fekete, Anna Lubiw |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
Pages | 35.1-35.10 |
ISBN (Electronic) | 9783959770095 |
DOIs | |
State | Published - 1 Jun 2016 |
Event | 32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States Duration: 14 Jun 2016 → 17 Jun 2016 |
Publication series
Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 51 |
ISSN (Print) | 1868-8969 |
Conference
Conference | 32nd International Symposium on Computational Geometry, SoCG 2016 |
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Country/Territory | United States |
City | Boston |
Period | 14/06/16 → 17/06/16 |
Bibliographical note
Publisher Copyright:© Dominic Dotterrer, Tali Kaufman, and Uli Wagner.
Funding
Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).
Funders | Funder number |
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Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung | SNSF-PP00P2-138948 |
Keywords
- Combinatorial topology
- Higher-dimensional expanders
- Selection Lemmas