Volume 16, issue 2 (2012)

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Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Tian-Jun Li and Weiwei Wu

Geometry & Topology 16 (2012) 1121–1169
Abstract

Given a Lagrangian sphere in a symplectic $4$–manifold $\left(M,\omega \right)$ with ${b}^{+}=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $\kappa$ of $\left(M,\omega \right)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans’ Hamiltonian uniqueness in the monotone case. On the existence side, when $\kappa =-\infty$, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

Keywords
Lagrangian sphere, symplectomorphism group
Mathematical Subject Classification 2010
Primary: 53D05, 53D12, 53D42
Publication
Received: 30 September 2011
Revised: 15 February 2012
Accepted: 3 March 2012
Published: 8 June 2012
Proposed: Ronald Fintushel
Seconded: Ronald J Stern, Leonid Polterovich
Authors
 Tian-Jun Li School of Mathematics University of Minnesota 206 Church Street Minneapolis MN 55455 USA http://www.math.umn.edu/~tjli Weiwei Wu School of Mathematics University of Minnesota 206 Church Street Minneapolis MN 55455 USA http://www.math.umn.edu/~wuxxx347