Wrinkled fibrations on near-symplectic manifolds

TL;DR

This paper explores the geometry of near-symplectic 4-manifolds and introduces a set of moves to relate different broken fibrations, using wrinkled fibrations and classical singularity theory, with applications to zero-set modifications.

Contribution

It introduces four moves for transforming broken fibrations on near-symplectic manifolds and analyzes their effects on the geometry, extending the theory of wrinkled fibrations.

Findings

Four moves connect any two deformation-equivalent broken fibrations.

Disproved Gay and Kirby's conjecture on achiral broken Lefschetz fibrations.

Demonstrated merging of zero-set components of near-symplectic forms.

Abstract

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow us to pass from any given fibration to any other broken fibration which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory.Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.