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Nearby Lagrangian fibers and Whitney sphere links

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Ekholm, Tobias 
Smith, Ivan 

Abstract

jats:pLet jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline1" />jats:tex-mathMisplaced &n&gt;3n&gt;3</jats:tex-math></jats:alternatives></jats:inline-formula>, and let jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline2" />jats:tex-mathL</jats:tex-math></jats:alternatives></jats:inline-formula> be a Lagrangian embedding of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline3" />jats:tex-mathRn</jats:tex-math></jats:alternatives></jats:inline-formula> into the cotangent bundle jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline4" />jats:tex-mathTRn</jats:tex-math></jats:alternatives></jats:inline-formula> of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline5" />jats:tex-mathRn</jats:tex-math></jats:alternatives></jats:inline-formula> that agrees with the cotangent fiber jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline6" />jats:tex-mathTxRn</jats:tex-math></jats:alternatives></jats:inline-formula> over a point jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline7" />jats:tex-mathx≠0</jats:tex-math></jats:alternatives></jats:inline-formula> outside a compact set. Assume that jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline8" />jats:tex-mathL</jats:tex-math></jats:alternatives></jats:inline-formula> is disjoint from the cotangent fiber at the origin. The projection of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline9" />jats:tex-mathL</jats:tex-math></jats:alternatives></jats:inline-formula> to the base extends to a map of the jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline10" />jats:tex-mathn</jats:tex-math></jats:alternatives></jats:inline-formula>-sphere jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline11" />jats:tex-mathSn</jats:tex-math></jats:alternatives></jats:inline-formula> into jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline12" />jats:tex-mathRn∖{0}</jats:tex-math></jats:alternatives></jats:inline-formula>. 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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X17007692_inline13" />jats:tex-mathTRn</jats:tex-math></jats:alternatives></jats:inline-formula>, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [jats:italicExact Lagrangian immersions with a single double point</jats:italic>, J. Amer. Math. Soc. jats:bold29</jats:bold> (2016), 1–59] and Ekholm and Smith [jats:italicExact Lagrangian immersions with one double point revisited</jats:italic>, Math. Ann. jats:bold358</jats:bold> (2014), 195–240] with symplectic field theory.</jats:p>

Description

Keywords

Lagrangian link, Whitney sphere, Floer equation, moduli space of holomorphic disks, symplectic field theory, Pontrjagin-Thom construction, stable homotopy groups of spheres

Journal Title

COMPOSITIO MATHEMATICA

Conference Name

Journal ISSN

0010-437X
1570-5846

Volume Title

154

Publisher

Wiley
Sponsorship
Engineering and Physical Sciences Research Council (EP/N01815X/1)