Indefinite Morse 2-functions; broken fibrations and generalizations

TL;DR

This paper studies indefinite Morse 2-functions, a generalization of broken fibrations, proving existence and uniqueness results for maps from manifolds to surfaces, with implications for understanding their topological structure.

Contribution

It establishes existence and uniqueness theorems for indefinite Morse 2-functions mapping to arbitrary surfaces, extending previous results to connected fibers.

Findings

Existence of indefinite Morse 2-functions for arbitrary compact, oriented surfaces.

Uniqueness up to a set of moves for homotopic indefinite Morse 2-functions.

Extension of results to functions with connected fibers.

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.