## Geometry & Topology

### Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

#### Abstract

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

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

Dates
Revised: 15 February 2012
Accepted: 3 March 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732415

Digital Object Identifier
doi:10.2140/gt.2012.16.1121

Mathematical Reviews number (MathSciNet)
MR2946805

Zentralblatt MATH identifier
1253.53073

#### Citation

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. https://projecteuclid.org/euclid.gt/1513732415

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