Geometry & Topology

Constructing Lefschetz-type fibrations on four-manifolds

David T Gay and Robion Kirby

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We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4–manifolds. These are generalizations of Lefschetz fibrations over the 2–sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1–handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4–manifolds without 3– and 4–handles, such as the homotopy 4–spheres arising from nontrivial balanced presentations of the trivial group.

Article information

Geom. Topol., Volume 11, Number 4 (2007), 2075-2115.

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
Secondary: 57R17: Symplectic and contact topology

Lefschetz fibrations round handles open book decompositions Andrews–Curtis conjecture Gluck construction achiral near-symplectic forms


Gay, David T; Kirby, Robion. Constructing Lefschetz-type fibrations on four-manifolds. Geom. Topol. 11 (2007), no. 4, 2075--2115. doi:10.2140/gt.2007.11.2075.

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