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Nearby Lagrangian fibers and Whitney sphere links
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. nst Mittag Leffler, Aurav 17, S-18260 Djursholm, Sweden..
Univ Cambridge, Ctr Math Sci, Wilberforce Rd, Cambridge CB3 0WB, England..
2018 (English)In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 154, no 4, p. 685-718Article in journal (Refereed) Published
Abstract [en]

Let n > 3, and let L be a Lagrangian embedding of R-n into the cotangent bundle T*R-n of R-n that agrees with the cotangent fiber T*R-x(n) over a point x not equal 0 outside a compact set. Assume that L is disjoint from the cotangent fiber at the origin. The projection of L to the base extends to a map of the n-sphere S-n into R-n\{0}. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in T*R-n, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1-59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195-240] with symplectic field theory.

Place, publisher, year, edition, pages
CAMBRIDGE UNIV PRESS , 2018. Vol. 154, no 4, p. 685-718
Keywords [en]
Lagrangian link, Whitney sphere, Floer equation, moduli space of holomorphic disks, symplectic field theory, Pontrjagin-Thom construction, stable homotopy groups of spheres
National Category
Geometry
Identifiers
URN: urn:nbn:se:uu:diva-351565DOI: 10.1112/S0010437X17007692ISI: 000427957900002OAI: oai:DiVA.org:uu-351565DiVA, id: diva2:1210938
Funder
Swedish Research CouncilKnut and Alice Wallenberg FoundationAvailable from: 2018-05-30 Created: 2018-05-30 Last updated: 2018-05-30Bibliographically approved

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