Singular fibrations over surfaces

TL;DR

This paper explores the theory of singular fibrations over surfaces, providing new constructions, connecting classical results, and establishing non-existence results for certain 4-manifolds.

Contribution

It introduces methods to construct singular fibrations with a single singularity and proves non-existence of such fibrations on certain 4-manifolds.

Findings

Constructed examples of singular fibrations with one singularity

Connected classical results on 2-plane fields to singular fibrations

Proved non-existence of singular fibrations on 4-manifolds with large first Betti number and zero second Betti number

Abstract

Singular fibrations generalize achiral Lefschetz fibrations of 4-manifolds over surfaces while sharing some of their properties. For instance, relatively minimal singular fibrations are determined by their monodromy. We explain how to construct examples of singular fibrations with a single singularity and Matsumoto's construction of singular fibrations of the sphere $S^4$. Previous results of Hirzebruch and Hopf on 2-plane fields with finitely many singularities are outlined in connection with the work of Neumann and Rudolph on the Hopf invariant. Eventually, we prove that closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.