Geometry & Topology

Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Tian-Jun Li and Weiwei Wu

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Given a Lagrangian sphere in a symplectic 4–manifold (M,ω) with b+=1, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension κ of (M,ω) is , 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 κ=, 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.

Article information

Geom. Topol., Volume 16, Number 2 (2012), 1121-1169.

Received: 30 September 2011
Revised: 15 February 2012
Accepted: 3 March 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D05: Symplectic manifolds, general 53D12: Lagrangian submanifolds; Maslov index 53D42: Symplectic field theory; contact homology

Lagrangian sphere symplectomorphism group


Li, Tian-Jun; Wu, Weiwei. Lagrangian spheres, symplectic surfaces and the symplectic mapping class group. Geom. Topol. 16 (2012), no. 2, 1121--1169. doi:10.2140/gt.2012.16.1121.

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