Existence of broken Lefschetz fibrations

Details Existence of broken Lefschetz fibrations Existence of broken Lefschetz fibrations

2008-01-21

arxiv.org/abs/0801.3139v2

Mathematics

Geometric Topology

11 pages, 5 figures. Remark 3.4 is updated, and an example of a broken Lefschetz fibration on the complex projective plane is

Scientific paper

We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber. Moreover, we can arrange it so that there is only one Lefschetz critical point when the Euler characteristic e(X) is odd, and none when e(X) is even. We make use of topological modifications of smooth maps with fold and cusp singularities due to Saeki and Levine, and thus we get alternative proofs of previous existence results. Also shown is the existence of broken Lefschetz pencils with connected fibers on any near-symplectic 4-manifold.

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