Constructing Lefschetz-type fibrations on four-manifolds

Abstract

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4-manifolds without 3- and 4-handles, such as the homotopy 4-spheres arising from nontrivial balanced presentations of the trivial group.

42 pages, 13 figures

References in corpus (6)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Lagrangian matching invariants for fibred four-manifolds: II
• Lagrangian matching invariants for fibred four-manifolds: I
• Kahler decomposition of 4-manifolds
• Pseudoholomorphic punctured spheres in R x (S^1 x S^2) : Properties and existence
• Pseudoholomorphic punctured spheres in R x (S^1 x S^2) : Moduli space parametrizations

Cited by in corpus (27)

• Wrinkled fibrations on near-symplectic manifolds
• Lagrangian matching invariants for fibred four-manifolds: I
• The h-principle for broken Lefschetz fibrations
• Every 4-Manifold is BLF
• Simplifying indefinite fibrations on 4-manifolds
• Fiber connected, indefinite Morse 2-functions on connected n-manifolds
• Fibrations and log-symplectic structures
• Cohomotopy sets of 4-manifolds
• Indefinite Morse 2-functions; broken fibrations and generalizations
• Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds
• Existence of broken Lefschetz fibrations
• Modification rule of monodromies in R_2-move
• Classification of broken Lefschetz fibrations with small fiber genera
• Open book decompositions of fibre sums in contact topology
• Braiding link cobordisms and non-ribbon surfaces
• Broken Lefschetz fibrations and smooth structures on 4-manifolds
• Broken Lefschetz fibrations and mapping class groups
• Fibrations in semi-toric and generalized complex geometry
• Near-symplectic 2n-manifolds
• Existence of 2-parameter crossings, with applications
• On deformations of isolated singularities of polar weighted homogeneous mixed polynomials
• Handlebody argument for modifying achiral Lefschetz singularities
• Maps on 3-manifolds given by surgery
• Poisson structures of near-symplectic manifolds and their cohomology
• Universal Lefschetz fibrations and Lefschetz cobordisms
• (Achiral) Lefschetz fibration embeddings of $4$-manifolds
• Constructing a broken Lefschetz fibration of S^4 with a spun or twist-spun torus knot fiber