Spherical Lagrangians via ball packings and symple

Authors: Matthew Strom Borman TianJun Li Weiwei Wu

Publish Date: 2013/03/12

Volume: 20, Issue: 1, Pages: 261-283

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Abstract

In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian S2 or mathbbRP 2 in symplectic manifolds that are rational or ruled Via a symplectic cutting construction this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions smooth knotting and unknottedness results for spherical Lagrangians the transitivity of the action of the symplectic Torelli group classifying Lagrangian isotopy classes in the presence of knotting and detecting Floertheoretically essential Lagrangian tori in the del Pezzo surfacesThe authors warmly thank Selman Akbulut Josef Dorfmeister Ronald Fintushel Robert Gompf Dusa McDuff and Leonid Polterovich for their interest in this work and many helpful correspondences Particular thanks are due to Dusa McDuff for generously sharing early versions of her paper 36 with us which plays a key role in our arguments We would also like to thank the anonymous referee for valuable comments suggestions clarifications and pointing us to the fact that Corollary 121 leads to a description of the Hamiltonian isotopy classes of Lagrangian spheres in a compact symplectic manifold where there are Hamiltonian knotted Lagrangian spheres Matthew Strom Borman was partially supported by NSFgrant DMS 1006610 TianJun Li and Weiwei Wu were supported by NSFgrant DMS 0244663We first recall from 28 Definition 33 that a stable spherical symplectic configuration is an ordered configuration of symplectic spheres with the following properties 1 c 1ge 1 for all irreducible components 2 the intersection numbers between two different components are 0 or 1 3 they are simultaneously holomorphic with respect to some almost complex structure J tamed by the symplectic form We will call them stable configurations for brevity In the proof of 28 Theorem 15 the following intermediate result is reachedIn mathbbCP 24overlinemathbbCP 2 let L 1 and L 2 be Lagrangian spheres in the homology class E 1 E 2 and suppose they are disjoint from a stable configuration with irreducible components in classes HE 1E 2 HE 3E 4 E 3 E 4 then L 1 and L 2 are Hamiltonian isotopic in the complement of the stable configurationIn particular in the proof of 28 Theorem 15 one uses 28 Proposition 68 to show that L 1 and L 2 are Hamiltonian isotopic in the complement of the stable configuration The same holds true for mathbbCP 2k+1overlinemathbbCP 2 as well for k=12 with the stable configurations specified in 28Fix a Darboux chart U psubset M that is disjoint from L 1 cup L 2 and centered at the point p in M By blowing up a symplectically embedded ball B psubset U p we can build a symplectic manifold Mprime =mathbbCP 24overlinemathbbCP 2omega prime with a exceptional sphere C such that H 2Mprime mathbbZ has a basis identified with the union of a basis of H 2MmathbbZ and C the intersection product L i cap C = 0

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