Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic
–manifolds, ie, manifolds
equipped with a closed –form
which is symplectic outside a union of embedded
–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.