#### Volume 19, issue 5 (2015)

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### David T Gay and Robion Kirby

Geometry & Topology 19 (2015) 2465–2534
##### Abstract

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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Primary: 57M50
Secondary: 57R17