Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Tian Jun Li, Weiwei Wu

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

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 inter section 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.

Original languageEnglish (US)
Pages (from-to)1121-1169
Number of pages49
JournalGeometry and Topology
Volume16
Issue number2
DOIs
StatePublished - 2012

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