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 Gromovâ€™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 (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the dcells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any ddimensional piecewiselinear 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:

15729168

Publisher:

Springer

Date Published:

20171104

Start Page: 
Epub ahead of print

Copyright Statement: 
CC BY 4.0

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

URL: 

DOI: 
10.1007/s1071101702914

Open access: 
yes (OA journal) 