On expansion and topological overlap

Dominic Dotterrer, Tali Kaufman, Uli Wagner

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations


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 languageEnglish
Title of host publication32nd International Symposium on Computational Geometry, SoCG 2016
EditorsSandor Fekete, Anna Lubiw
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770095
StatePublished - 1 Jun 2016
Event32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States
Duration: 14 Jun 201617 Jun 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference32nd International Symposium on Computational Geometry, SoCG 2016
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© Dominic Dotterrer, Tali Kaufman, and Uli Wagner.


Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).

FundersFunder number
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen ForschungSNSF-PP00P2-138948


    • Combinatorial topology
    • Higher-dimensional expanders
    • Selection Lemmas


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