License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.35
URN: urn:nbn:de:0030-drops-59270
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Dotterrer, Dominic ; Kaufman, Tali ; Wagner, Uli

On Expansion and Topological Overlap

LIPIcs-SoCG-2016-35.pdf (0.5 MB)


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 -> R^d there exists a point p in R^d whose preimage intersects a positive fraction mu > 0 of the d-cells of X. More generally, the conclusion holds if R^d is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant \mu that depends only on d and on the expansion properties of X, but not on M.

BibTeX - Entry

  author =	{Dominic Dotterrer and Tali Kaufman and Uli Wagner},
  title =	{{On Expansion and Topological Overlap}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{35:1--35:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-59270},
  doi =		{10.4230/LIPIcs.SoCG.2016.35},
  annote =	{Keywords: Combinatorial Topology, Selection Lemmas, Higher-Dimensional Expanders}

Keywords: Combinatorial Topology, Selection Lemmas, Higher-Dimensional Expanders
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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