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

Mikhail Gromov

doi:10.1007/s00039-009-0021-7

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

Mikhail Gromov

Mathematics

Research output: Contribution to journal › Article › peer-review

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	https://doi.org/10.1007/s00039-009-0021-7
State	Published - 2009

Keywords

Expanders
Hyperbolic manifolds
Morse theory
Simplicial volume
Singularities

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.1007/s00039-009-0021-7

Cite this

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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.",

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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

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JF - Geometric and Functional Analysis

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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