## 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 manifold M, with a constant μ that depends only on d and on the expansion properties of X, but not on M.

Original language | English |
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Pages (from-to) | 307-317 |

Number of pages | 11 |

Journal | Geometriae Dedicata |

Volume | 195 |

Issue number | 1 |

DOIs | |

State | Published - 1 Aug 2018 |

### Bibliographical note

Publisher Copyright:© 2017, The Author(s).

### Funding

Open access funding provided by Institute of Science and Technology (IST Austria). We would like to thank the anonymous referees for many helpful remarks concerning the presentation. Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948). An extended abstract of this paper [5] appeared in the Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016).

Funders | Funder number |
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Institute of Science and Technology | |

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung | SNSF-PP00P2-138948 |

Instituto Superior Técnico |

## Keywords

- Cell complexes
- Expansion
- High dimensional expansion
- Topological overlapping