#### Volume 14, issue 2 (2010)

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The $h$–principle for broken Lefschetz fibrations

### Jonathan Williams

Geometry & Topology 14 (2010) 1015–1061
##### Abstract

It is known that an arbitrary smooth, oriented $4$–manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken Lefschetz fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken Lefschetz fibration in a given homotopy class of smooth maps. One notable application is that adding an additional “projection" move generates all broken Lefschetz fibrations, regardless of homotopy class. The paper ends with further applications and open problems.

##### Keywords
broken, Lefschetz fibration, $4$–manifold, stable map
##### Mathematical Subject Classification 2000
Primary: 57M50, 57N13
Secondary: 57R70, 57R17