Geometry & Topology

Indefinite Morse $2$–functions: Broken fibrations and generalizations

David T Gay and Robion Kirby

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A Morse 2–function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2–function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2–functions mapping to arbitrary compact, oriented surfaces. “Uniqueness” means there is a set of moves which are sufficient to go between two homotopic indefinite Morse 2–functions while remaining indefinite throughout. We extend the existence and uniqueness results to indefinite, Morse 2–functions with connected fibers.

Article information

Geom. Topol., Volume 19, Number 5 (2015), 2465-2534.

Received: 3 February 2011
Revised: 3 February 2011
Accepted: 17 November 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57R17: Symplectic and contact topology

broken fibration Morse function Cerf theory definite fold elliptic umbilic


Gay, David T; Kirby, Robion. Indefinite Morse $2$–functions: Broken fibrations and generalizations. Geom. Topol. 19 (2015), no. 5, 2465--2534. doi:10.2140/gt.2015.19.2465.

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