On expansion and topological overlap

Dotterrer, Dominic and Kaufman, Tali and Wagner, Uli (2016) On expansion and topological overlap. In: SoCG: Symposium on Computational Geometry, June 14 - 17, 2016, Boston, MA, USA.

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Official URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.35


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.

Item Type: Conference or Workshop Item (Paper)
Uncontrolled Keywords: Combinatorial topology, Higher-dimensional expanders, Selection Lemmas
Subjects: 500 Science > 510 Mathematics
Research Group: Wagner Group
SWORD Depositor: Sword Import User
Depositing User: Sword Import User
Date Deposited: 21 Jul 2016 07:15
Last Modified: 30 Aug 2017 14:42
URI: https://repository.ist.ac.at/id/eprint/623

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