### Abstract

We find lower bounds on the topological complexity of the critical (values) sets Σ(F)⊂Y of generic smooth maps F: X→Y, as well as on the complexity of the fibers F^{-1} (y)⊂ X in terms of the topology of X and Y, where the relevant topological invariants of X are often encoded in the geometry of some Riemannian metric supported by X.

Original language | English (US) |
---|---|

Pages (from-to) | 743-841 |

Number of pages | 99 |

Journal | Geometric and Functional Analysis |

Volume | 19 |

Issue number | 3 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### Keywords

- Expanders
- Hyperbolic manifolds
- Morse theory
- Simplicial volume
- Singularities

### ASJC Scopus subject areas

- Geometry and Topology
- Analysis

### Cite this

**Singularities, expanders and topology of maps. Part 1 : Homology versus volume in the spaces of cycles.** / Gromov, Mikhael.

Research output: Contribution to journal › Article

*Geometric and Functional Analysis*, vol. 19, no. 3, pp. 743-841. https://doi.org/10.1007/s00039-009-0021-7

}

TY - JOUR

T1 - Singularities, expanders and topology of maps. Part 1

T2 - Homology versus volume in the spaces of cycles

AU - Gromov, Mikhael

PY - 2009

Y1 - 2009

N2 - We find lower bounds on the topological complexity of the critical (values) sets Σ(F)⊂Y of generic smooth maps F: X→Y, as well as on the complexity of the fibers F-1 (y)⊂ X in terms of the topology of X and Y, where the relevant topological invariants of X are often encoded in the geometry of some Riemannian metric supported by X.

AB - We find lower bounds on the topological complexity of the critical (values) sets Σ(F)⊂Y of generic smooth maps F: X→Y, as well as on the complexity of the fibers F-1 (y)⊂ X in terms of the topology of X and Y, where the relevant topological invariants of X are often encoded in the geometry of some Riemannian metric supported by X.

KW - Expanders

KW - Hyperbolic manifolds

KW - Morse theory

KW - Simplicial volume

KW - Singularities

UR - http://www.scopus.com/inward/record.url?scp=78649402676&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649402676&partnerID=8YFLogxK

U2 - 10.1007/s00039-009-0021-7

DO - 10.1007/s00039-009-0021-7

M3 - Article

VL - 19

SP - 743

EP - 841

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 3

ER -