# Classification of boundary Lefschetz fibrations over the disc

## TL;DR

This paper characterizes four-manifolds that admit boundary Lefschetz fibrations over the disc, linking them to specific connected sums and stable generalized complex structures with a single type change component.

## Contribution

It provides a complete classification of four-manifolds with boundary Lefschetz fibrations over the disc and relates these to stable generalized complex structures.

## Key findings

- Manifolds admitting boundary Lefschetz fibrations are classified as specific connected sums.
- These manifolds are exactly those with stable structures having a single type change component.
- The classification links boundary Lefschetz fibrations to stable generalized complex structures.

## Abstract

We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\# m\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ or $\# m (S^2 \times S^2)$. Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, we conclude that the manifolds $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\#(2m+1)\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ and $\# (2m+1) S^2 \times S^2$ admit stable structures whose type change locus has a single component and are the only four-manifolds whose stable structure arise from boundary Lefschetz fibrations over the disc.