Geometry & Topology

The $h$–principle for broken Lefschetz fibrations

Jonathan Williams

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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

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

Received: 1 July 2009
Accepted: 23 February 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]
Secondary: 57R70: Critical points and critical submanifolds 57R17: Symplectic and contact topology

broken Lefschetz fibration $4$–manifold stable map


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

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