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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.

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.

Reviewer: Jarek Kedra (Aberdeen)

### 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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