## Geometry & Topology

### The $h$–principle for broken Lefschetz fibrations

Jonathan Williams

#### 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.

#### Article information

Source
Geom. Topol., Volume 14, Number 2 (2010), 1015-1061.

Dates
Accepted: 23 February 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732211

Digital Object Identifier
doi:10.2140/gt.2010.14.1015

Mathematical Reviews number (MathSciNet)
MR2629899

Zentralblatt MATH identifier
1204.57027

#### Citation

Williams, Jonathan. The $h$–principle for broken Lefschetz fibrations. Geom. Topol. 14 (2010), no. 2, 1015--1061. doi:10.2140/gt.2010.14.1015. https://projecteuclid.org/euclid.gt/1513732211

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