Geometry & Topology 16 (2012) 1121–1169

DOI: 10.2140/gt.2012.16.1121

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.

Keywords

Lagrangian sphere, symplectomorphism group

Mathematical Subject Classification

Primary: 53D05, 53D12, 53D42

References

Publication

Received: 30 September 2011
Revised: 15 February 2012
Accepted: 3 March 2012
Published: 8 June 2012
Proposed: Ronald Fintushel
Seconded: Ronald J Stern, Leonid Polterovich

Authors