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

David T Gay and Robion Kirby

Geometry & Topology 19 (2015) 2465–2534

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.

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broken fibration, Morse function, Cerf theory, definite fold, elliptic umbilic
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 57R17
Received: 3 February 2011
Revised: 3 February 2011
Accepted: 17 November 2014
Published: 20 October 2015
Proposed: Simon Donaldson
Seconded: David Gabai, Cameron Gordon
David T Gay
Euclid Lab
160 Milledge Terrace
Athens, GA 30606
Robion Kirby
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840