Trisections of Lefschetz pencils

Gay, David

doi:10.2140/agt.2016.16.3523

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Trisections of Lefschetz pencils

David T Gay

Algebraic & Geometric Topology 16 (2016) 3523–3531

DOI: 10.2140/agt.2016.16.3523

Abstract

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.

Keywords

Lefschetz pencil, symplectic, 4-manifold, trisection, open book

Mathematical Subject Classification 2010

Primary: 57M99, 57M50

Secondary: 57R45, 57R65, 57R17

References

Publication

Received: 30 October 2015

Revised: 10 May 2016

Accepted: 19 May 2016

Published: 15 December 2016

Authors

David T Gay
Euclid Lab 160 Milledge Terrace Athens, GA 30606 United States	Department of Mathematics University of Georgia Athens, GA 30602 United States
http://euclidlab.org/david-gay/

Volume 16, issue 6 (2016)