Nearby Lagrangian fibers and Whitney sphere links
arXiv:1609.07591 · doi:10.1112/S0010437X17007692
Abstract
Let n>3, and let L be a Lagrangian embedding of an n-disk into the cotangent bundle of n-dimensional Euclidean space that agrees with the cotangent fiber over a non-zero point x 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 into the complement of the origin in Euclidean n-space . We show that this map is homotopically trivial, answering a question of Y. 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 2n-dimensional space, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from the authors' previous work, constructing bounding manifolds from moduli spaces of Floer-holomorphic disks, with symplectic field theory.
v2: 39 pages, 2 figures. Numerous minor corrections and clarifications to take account of referees' suggestions. This version to appear in Compositio