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Surgery and Invariants of Lagrangian Surfaces
Surgery and Invariants of Lagrangian Surfaces
We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists as constructed by Paul Seidel. We also constructed a new symplectic invariant, called y-index, for orientable closed Lagrangian surfaces immersed in a parallelizable symplectic 4-manifold W. With y-index we proved that L and L' are not Hamiltonian isotopic. We also obtained new examples of nullhomologous Lagrangian tori which are smooth isotopic but not Hamiltonian isotopic.
Version 3: Typos and minor mistakes from Sections 1.1-3.3 corrected. Sections 3.4-5.4 revised, definition of a $p$-curve modified, 3 figures added. 90 pages
arXiv: Mathematics::Symplectic Geometry Mathematics::Geometric Topology
Mathematics - Symplectic Geometry, Mathematics - Differential Geometry, Mathematics - Geometric Topology, Symplectic Geometry (math.SG), Differential Geometry (math.DG), Geometric Topology (math.GT), FOS: Mathematics
Mathematics - Symplectic Geometry, Mathematics - Differential Geometry, Mathematics - Geometric Topology, Symplectic Geometry (math.SG), Differential Geometry (math.DG), Geometric Topology (math.GT), FOS: Mathematics
arXiv: Mathematics::Symplectic Geometry Mathematics::Geometric Topology
5 Examples 64 5.1 Lagrangian surfaces in R4 . . . . . . . . . . . . . . . . . . . . 64 5.1.1 S1-invariant p-domain . . . . . . . . . . . . . . . . . . 64 5.1.2 Whitney sphere and tori . . . . . . . . . . . . . . . . . 67 5.1.3 Fibers of an integrable system . . . . . . . . . . . . . 69 5.2 Sphere in T S2 . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Surfaces in the plumbing of cotangent bundles . . . . . . . . . 79 5.4 Monotone tori in Wn . . . . . . . . . . . . . . . . . . . . . . . 86 Proof. For 0 i n pick an open neighborhood Ui of Di such that closures of Ui are pairwise disjoint and for each i, L \ Ui is an open annulus. In each Ui we construct a pair embedded Lagrangian disks D0; Di00 as before, so that i
interiors of D0; Di00 are disjoint from L, both disks are tangent to L along i
their corresponding boundaries Ci0; Ci00, and Di0; Di00 intersect transversally , and
[13] A. Knapp, Lagrangian tori in closed 4-manifolds, J. Topol. 3 (2010), no. 2, 333342.
[14] D. McDu and D. Salamon, Introduction to Symplectic Topology, 2nd ed., Oxford Mathematical Monographs, 1998.
[15] Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks, I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949993.
[16] Y.-G. Oh, Addendum to "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I", Comm. Pure Appl. Math. 48 (1995), no. 11, 12991302.
[17] L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198{210.
[18] M. Pozniak, Floer homology, Novikov rings and clean intersections. Northern California Symplectic Geometry Seminar, 119{181, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999. [OpenAIRE]
[19] P. Seidel, Lagrangian two-spheres can be symplectically knotted, J. Di . Geom. 52 (1999) no. 1, 145{171.
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We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists as constructed by Paul Seidel. We also constructed a new symplectic invariant, called y-index, for orientable closed Lagrangian surfaces immersed in a parallelizable symplectic 4-manifold W. With y-index we proved that L and L' are not Hamiltonian isotopic. We also obtained new examples of nullhomologous Lagrangian tori which are smooth isotopic but not Hamiltonian isotopic.
Version 3: Typos and minor mistakes from Sections 1.1-3.3 corrected. Sections 3.4-5.4 revised, definition of a $p$-curve modified, 3 figures added. 90 pages