Volume 13, issue 1 (2009)

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Wrinkled fibrations on near-symplectic manifolds

Yankı Lekili

Appendix: R İnanç Baykur

Geometry & Topology 13 (2009) 277–318

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4–manifolds, ie, manifolds equipped with a closed 2–form which is symplectic outside a union of embedded 1–dimensional submanifolds, and broken Lefschetz fibrations on them; see Auroux, Donaldson and Katzarkov [?] and Gay and Kirby [?]. We present a set of four moves which allow us to pass from any given broken fibration to any other which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory. Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.

broken Lefschetz fibration, wrinkled fibration, near-symplectic, manifold
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57R17, 57R45
Received: 12 February 2008
Revised: 17 August 2008
Accepted: 26 July 2008
Preview posted: 5 November 2008
Published: 1 January 2009
Proposed: Rob Kirby
Seconded: Ron Fintushel, Simon Donaldson
Yankı Lekili
Department of Mathematics
Cambridge MA 02139
R İnanç Baykur
Department of Mathematics
Brandeis University
Waltham, MA 02454