Nearby Lagrangian fibers and Whitney sphere links

TL;DR

This paper proves that certain Lagrangian embeddings in cotangent bundles are homotopically trivial when projected, extending to constraints on Lagrangian disks and links, using Floer theory and symplectic field theory.

Contribution

It establishes homotopical triviality of projections of specific Lagrangian embeddings and generalizes this to constraints on Lagrangian disks and sphere links.

Findings

Projection maps are homotopically trivial under given conditions

Constraints on embedded Lagrangian disks in complements of other Lagrangians

Restrictions on Lagrangian sphere links with double points

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.