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**Constructing Lefschetz-type fibrations on four-manifolds.**
*(English)*
Zbl 1135.57009

In 1999, S. Donaldson discovered that symplectic 4-manifolds have Lefschetz pencil structures. In 2005, Auroux, Donaldson and Katzarkov showed that a near-symplectic 4-manifold is a broken Lefschetz fibration (BLF). In 2006, Etnyre and Fuller showed that a 4-manifold connected sum with a 2-sphere bundle over \(S^2\) is an achiral Lefschetz fibration (ALF). Let \(X\) be an arbitrary closed 4-manifold and \(F\) be a closed surface in \(X\) with self-intersection \(F\cdot F= 0\). Then the authors show that there is a broken, achiral Lefschetz fibration (BALF) from \(X\) to \(S^2\) with \(F\) as a fiber. To prove the result they take the concave BLF on \(F\times B^2\) and add enough round 1-handles so that the complement is built with \(0\)-, \(1\)- and \(2\)-handles. This induces an open book decomposition (OBD) on the boundary of this concave piece. Then they construct a convex BLF on the complement, inducing an OBD on its boundary which supports a contact structure homotopic to the contact structure supported by the OBD coming from the concave piece. They arrange that both contact structures are overtwisted and so isotopic. By Giroux’s work on open books the two OBDs have a common positive stabilization. The authors arrange concave boundaries and convex boundaries so that the open books match. They give a list of questions in the last section.

Reviewer: Yong Seung Cho (Seoul)

### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

57R65 | Surgery and handlebodies |

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\textit{D. T. Gay} and \textit{R. Kirby}, Geom. Topol. 11, 2075--2115 (2007; Zbl 1135.57009)

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