Abstract
In this paper we classify Lagrangian spheres in \(A_n\)-surface singularities up to Hamiltonian isotopy. Combining with a result of Ritter (Geom Funct Anal 20(3):779–816, 2010), this yields a complete classification of exact Lagrangians in \(A_n\)-surface singularities. Our main new tool is the application of a technique which we call ball-swappings and its relative version.
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References
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Acknowledgments
The author is deeply indebted to Richard Hind, who inspired the idea in this paper during his stay in Michigan State University. I warmly thank Dusa McDuff for generously sharing preliminary drafts of her recent preprint with Emmanuel Opshtein [18] and for explaining many details, which are very helpful to the current paper. I also thank an anonymous referee for carefully checking many details, as well as suggesting an alternative approach of Lemma 3.4. Many ideas involved in this work were originally due to Jonny Evans in his excellent series of papers [5, 6], and the ball-swapping construction used here stems out from the beautiful ideas exhibited in Seidel’s lecture notes [26]. This work is partially motivated by a talk by Yanki Lekili in MSRI. I would also like to thank the Geometry/Topology group of Michigan State University for providing me a friendly and inspiring working environment, as well as supports for my visitors. The author is supported by NSF Focused Research Grants DMS-0244663.