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Wrinkled fibrations on near-symplectic manifolds. (English) Zbl 1164.57006

A broken fibration on a closed four manifold is a map to a closed surface with a finite set of Lefschetz singularities and a one-dimensional set of other (precisely defined) singularities. A wrinkled fibration on a closed four manifold is a map to a closed surface which is a broken fibration on the complement of a finite set of cusp singularities.
The main results of the paper state that:
(1) a wrinkled fibration is homotopic to a broken one by a homotopy supported near cusp singularities;
(2) a broken fibration is homotopic to a wrinkled fibration with no Lefschetz singularities by a homotopy supported near Lefschetz singularities.
The proof consists of a definition of certain basic moves and their careful analysis. As an application the author proves that an achiral broken Lefschetz fibration can be deformed into a broken Lefschetz fibration using the above moves. This disproves a conjecture of Gay and Kirby. The author also study the change of the near-symplectic geometry under the above moves.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
57R45 Singularities of differentiable mappings in differential topology
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References:

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