Singular Lefschetz pencils

Details Singular Lefschetz pencils Singular Lefschetz pencils

2004-10-14

arxiv.org/abs/math/0410332v2

Geom. Topol. 9(2005) 1043-1114

Mathematics

Differential Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper24.abs.html

Scientific paper

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X,omega) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S^1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form. Conversely, from such a decomposition one can recover a near-symplectic structure.

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