Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Tianjun Li University of Minnesota Weiwei Wu University of Minnesota

Symplectic Geometry mathscidoc:1705.34001

Geom. Topol., 16, (2), 1121–1169, 2012
Given a Lagrangian sphere in a symplectic 4–manifold $(M, \omega)$ with $b^{+}=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension  $\kappa$ of $(M,\omega)$ 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.
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@inproceedings{tianjun2012lagrangian,
  title={Lagrangian spheres, symplectic surfaces and the symplectic mapping class group},
  author={Tianjun Li, and Weiwei Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530150937103036766},
  booktitle={Geom. Topol.},
  volume={16},
  number={2},
  pages={1121–1169},
  year={2012},
}
Tianjun Li, and Weiwei Wu. Lagrangian spheres, symplectic surfaces and the symplectic mapping class group. 2012. Vol. 16. In Geom. Topol.. pp.1121–1169. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530150937103036766.
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