On genus-1 simplified broken Lefschetz fibrations

TL;DR

This paper classifies genus-1 simplified broken Lefschetz fibrations with fewer than six singularities, extending Kas's classification of genus-1 Lefschetz fibrations to a broader context involving near-symplectic 4-manifolds.

Contribution

It provides a complete classification of such fibrations with connected fibers and fewer than six singularities, and introduces new families conjectured to encompass all cases.

Findings

Classified diffeomorphism types of these fibrations.

Identified families of genus-1 simplified broken Lefschetz fibrations.

Extended Kas's theorem to broken Lefschetz fibrations.

Abstract

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas' classification theorem of genus-1 Lefschetz fibrations, which states that the total space of a non-trivial genus-1 Lefschetz fibration over $S^2$ is diffeomorphic to an elliptic surface E(n), for some $n\geq 1$.