On expansion and topological overlap Journal Article


Author(s): Dotterrer, Dominic; Kaufman, Tali; Wagner, Uli
Article Title: On expansion and topological overlap
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 (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the d-cells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any d-dimensional piecewise-linear manifold M, with a constant (Formula presented.) that depends only on d and on the expansion properties of X, but not on M.
Keywords: Cell complexes; Expansion; High dimensional expansion; Topological overlapping
Journal Title: Geometriae Dedicata
ISSN: 1572-9168
Publisher: Springer  
Date Published: 2017-11-04
Start Page: Epub ahead of print
Copyright Statement: CC BY 4.0
Sponsor: Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).
URL:
DOI: 10.1007/s10711-017-0291-4
Open access: yes (OA journal)
IST Austria Authors
  1. Uli Wagner
    50 Wagner
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