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

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.