## 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} whose preimage intersects 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.

Original language | English |
---|---|

Title of host publication | 32nd International Symposium on Computational Geometry, SoCG 2016 |

Editors | Sandor Fekete, Anna Lubiw |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 35.1-35.10 |

ISBN (Electronic) | 9783959770095 |

DOIs | |

State | Published - 1 Jun 2016 |

Event | 32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States Duration: 14 Jun 2016 → 17 Jun 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 51 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd International Symposium on Computational Geometry, SoCG 2016 |
---|---|

Country/Territory | United States |

City | Boston |

Period | 14/06/16 → 17/06/16 |

### Bibliographical note

Publisher Copyright:© Dominic Dotterrer, Tali Kaufman, and Uli Wagner.

### Funding

Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).

Funders | Funder number |
---|---|

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

## Keywords

- Combinatorial topology
- Higher-dimensional expanders
- Selection Lemmas