Indefinite Morse 2–functions : Broken fibrations and generalizations

Gay, David T¹ ; Kirby, Robion²

¹ Euclid Lab, 160 Milledge Terrace, Athens, GA 30606, USA
² Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-3840, USA

Geometry & topology, Tome 19 (2015) no. 5, pp. 2465-2534.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.2465

Classification : 57M50, 57R17
Keywords: broken fibration, Morse function, Cerf theory, definite fold, elliptic umbilic

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