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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 [Geom. Topol. 9
(2005) 1043--1114] and Gay and Kirby [Geom. Topol. 11
(2007) 2075--2115]. 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.
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