On expansion and topological overlap

Dominic Dotterrer, Tali Kaufman, Uli Wagner

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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 that is contained in the images of 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 manifold M, with a constant μ that depends only on d and on the expansion properties of X, but not on M.

Original languageEnglish
Pages (from-to)307-317
Number of pages11
JournalGeometriae Dedicata
Volume195
Issue number1
DOIs
StatePublished - 1 Aug 2018

Bibliographical note

Publisher Copyright:
© 2017, The Author(s).

Funding

Open access funding provided by Institute of Science and Technology (IST Austria). We would like to thank the anonymous referees for many helpful remarks concerning the presentation. Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948). An extended abstract of this paper [5] appeared in the Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016).

FundersFunder number
Institute of Science and Technology
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen ForschungSNSF-PP00P2-138948
Instituto Superior Técnico

    Keywords

    • Cell complexes
    • Expansion
    • High dimensional expansion
    • Topological overlapping

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