TY - JOUR
T1 - Lagrangian spheres, symplectic surfaces and the symplectic mapping class group
AU - Li, Tian Jun
AU - Wu, Weiwei
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
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U2 - 10.2140/gt.2012.16.1121
DO - 10.2140/gt.2012.16.1121
M3 - Article
AN - SCOPUS:84863479083
SN - 1465-3060
VL - 16
SP - 1121
EP - 1169
JO - Geometry and Topology
JF - Geometry and Topology
IS - 2
ER -