Generic fibrations around multiple fibers

TL;DR

This paper constructs explicit generic fibrations with only indefinite fold singularities around multiple fibers in 4-manifolds, enabling new explicit broken Lefschetz fibrations on elliptic surfaces and fibrations around Seifert fibered space fibers.

Contribution

It introduces a method to produce explicit generic fibrations around multiple fibers, extending fibrations to 4-manifolds with controlled singularities.

Findings

Constructed explicit generic fibrations around multiple fibers.

Provided explicit broken Lefschetz fibrations on all elliptic surfaces.

Produced generic fibrations around exceptional fibers of Seifert fibered spaces.

Abstract

Given some type of fibration on a 4-manifold $X$ with a torus regular fiber $T$, we may produce a new 4-manifold $X_T$ by performing torus surgery on $T$. There is a natural way to extend the fibration to $X_T$, but a multiple fiber (non-generic) singularity is introduced. We construct explicit generic fibrations (with only indefinite fold singularities) in a neighborhood of this multiple fiber. As an application this gives explicit constructions of broken Lefschetz fibrations on all elliptic surfaces (e.g. the family $E(n)_{p,q}$). As part of the construction we produce generic fibrations around exceptional fibers of Seifert fibered spaces.