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Trisections of Lefschetz pencils

David T Gay

Algebraic & Geometric Topology 16 (2016) 3523–3531

Donaldson [J. Differential Geom. 53 (1999) 205–236] showed that every closed symplectic 4–manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby [Geom. Topol. 20 (2016) 3097–3132] showed that every closed 4–manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pencil. This trisection is such that each of the three sectors is a regular neighborhood of a regular fiber of the pencil. This is a 4–dimensional analog of the following trivial 3–dimensional result: for every open book decomposition of a 3–manifold M, there is a decomposition of M into three handlebodies, each of which is a regular neighborhood of a page.

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Lefschetz pencil, symplectic, 4-manifold, trisection, open book
Mathematical Subject Classification 2010
Primary: 57M99, 57M50
Secondary: 57R45, 57R65, 57R17
Received: 30 October 2015
Revised: 10 May 2016
Accepted: 19 May 2016
Published: 15 December 2016
David T Gay
Euclid Lab
160 Milledge Terrace
Athens, GA 30606
United States
Department of Mathematics
University of Georgia
Athens, GA 30602
United States