On Expansion and Topological Overlap

2015/06/15 by Dominic Dotterrer, Dotterrer, Dominic, Tali Kaufman +3
Computer Science · Mathematics · #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Topology (math.GT) #cs.DM #math.GT

paper · pdf · doi:10.48550/arxiv.1506.04558

Minor revision, updated references

arxiv created 2016/09/17 · arxiv updated 2016/09/20

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 X→ R^d there exists a point p∈ R^d that is contained in the images of a positive fraction μ>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 μ that depends only on d and on the expansion properties of X, but not on M.

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