Geodesic
Trisections of Lefschetz pencils
Gay, David1
1Euclid Lab, 160 Milledge Terrace, Athens, GA 30606, United States, Department of Mathematics, University of Georgia, Athens, GA 30602, United States
Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3523-3531
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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.

DOI : 10.2140/agt.2016.16.3523
Classification : 57M99, 57M50, 57R45, 57R65, 57R17
Keywords: Lefschetz pencil, symplectic, 4-manifold, trisection, open book

Gay, David  1

1 Euclid Lab, 160 Milledge Terrace, Athens, GA 30606, United States, Department of Mathematics, University of Georgia, Athens, GA 30602, United States 
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Gay, David. Trisections of Lefschetz pencils. Algebraic and Geometric Topology, Tome 16 (2016) no. 6, pp. 3523-3531. doi: 10.2140/agt.2016.16.3523

[1] S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205

[2] D T Gay, R Kirby, Trisecting 4–manifolds, Geom. Topol. 20 (2016) 3097 | DOI

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