Wrinkled fibrations on near-symplectic manifolds

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.

35 pages, 12 figures. Final version. Minor corrections and clarifications

References in corpus (6)

• Lagrangian matching invariants for fibred four-manifolds: II
• Constructing Lefschetz-type fibrations on four-manifolds
• Lagrangian matching invariants for fibred four-manifolds: I
• Every 4-Manifold is BLF
• Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds
• Zero-sets of near-symplectic forms

Cited by in corpus (23)

• Trisecting 4-manifolds
• The h-principle for broken Lefschetz fibrations
• Every 4-Manifold is BLF
• Fiber connected, indefinite Morse 2-functions on connected n-manifolds
• On genus-1 simplified broken Lefschetz fibrations
• Indefinite Morse 2-functions; broken fibrations and generalizations
• Cohomotopy sets of 4-manifolds
• Elimination of cusps in dimension 4 and its applications
• Poisson structures on smooth 4-manifolds
• Existence of broken Lefschetz fibrations
• Geometric structures and Lie algebroids
• Modification rule of monodromies in R_2-move
• Poisson structures on wrinkled fibrations
• Braiding link cobordisms and non-ribbon surfaces
• Fibrations in semi-toric and generalized complex geometry
• Near-symplectic 2n-manifolds
• Existence of 2-parameter crossings, with applications
• Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6
• Handlebody argument for modifying achiral Lefschetz singularities
• The Kodaira Dimension of Lefschetz Fibrations
• Generic fibrations around multiple fibers
• Vertical 3-manifolds in simplified (2, 0)-trisections of 4-manifolds
• Poisson structures of near-symplectic manifolds and their cohomology