Constructing Lefschetz-type fibrations on four-manifolds

Gay, David T; Kirby, Robion

doi:10.2140/gt.2007.11.2075

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Constructing Lefschetz-type fibrations on four-manifolds

David T Gay and Robion Kirby

Geometry & Topology 11 (2007) 2075–2115

DOI: 10.2140/gt.2007.11.2075

arXiv: math.GT/0701084

Abstract

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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 57R17

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Publication

Published: 15 October 2007

Proposed: Simon Donaldson

Seconded: Ronald J Stern, Ron Fintushel

Authors

David T Gay
Department of Mathematics and Applied Mathematics University of Cape Town Private Bag X3 Rondebosch 7701 South Africa

Robion Kirby
University of California Berkeley CA 94720 USA

Volume 11, issue 4 (2007)