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 higherdimensional 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 dcells of X. More generally, the conclusion holds if ℝd is replaced by any ddimensional piecewiselinear (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; Higherdimensional expanders; Selection Lemmas

Conference Title:

SoCG: Symposium on Computational Geometry

Volume: 
51

Conference Dates:

June 14  17, 2016

Conference Location:

Boston, MA, USA

ISBN:

9781450325943

Publisher:

ACM

Date Published:

20160601

Start Page: 
35.1

End Page:

35.10

Copyright Statement: 
CCBY

Sponsor: 
Research supported by the Swiss National Science Foundation (Project SNSFPP00P2138948).

URL: 

DOI: 
10.4230/LIPIcs.SoCG.2016.35

Open access: 
yes (OA journal) 