Volume 11, issue 4 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Constructing Lefschetz-type fibrations on four-manifolds

David T Gay and Robion Kirby

Geometry & Topology 11 (2007) 2075–2115

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