On expansion and topological overlap Conference Paper

Author(s): Dotterrer, Dominic; Kaufman, Tali; Wagner, Uli
Title: On expansion and topological overlap
Title Series: LIPIcs
Affiliation IST Austria
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.
Keywords: Combinatorial topology; Higher-dimensional expanders; Selection Lemmas
Conference Title: SoCG: Symposium on Computational Geometry
Volume: 51
Conference Dates: June 14 - 17, 2016
Conference Location: Boston, MA, USA
ISBN: 978-1-4503-2594-3
Publisher: Schloss Dagstuhl- Leibniz-Zentrum für Informatik  
Date Published: 2016-06-01
Start Page: 35.1
End Page: 35.10
Copyright Statement: CC-BY
Sponsor: Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).
DOI: 10.4230/LIPIcs.SoCG.2016.35
Open access: yes (OA journal)
IST Austria Authors
  1. Uli Wagner
    43 Wagner
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