Every 4-Manifold is BLF

TL;DR

This paper proves that all compact smooth 4-manifolds admit a Broken Lefschetz Fibration structure, and those with positive second Betti number also admit a Broken Lefschetz Pencil, advancing the understanding of 4-manifold topology.

Contribution

It provides a topological proof that every compact smooth 4-manifold admits a BLF, improving previous results by Auroux, Donaldson, and Katzarkov.

Findings

All compact smooth 4-manifolds have a BLF structure.

4-manifolds with b_{2}^{+}> 0 have a BLF with nonempty base locus.

The proof uses handlebody theory in 4-dimensional topology.

Abstract

Here we show that every compact smooth 4-manifold X has a structure of a Broken Lefschetz Fibration (BLF in short). Furthermore, if b_{2}^{+}(X)> 0 then it also has a Broken Lefschetz Pencil structure (BLP) with nonempty base locus. This imroves a Theorem of Auroux, Donaldson and Katzarkov, and our proof is topological (i.e. uses 4-dimensional handlebody theory).