# Nearby Lagrangian fibers and Whitney sphere links

13 Jul 2018

Let \$n>3\$, and let \$L\$ be a Lagrangian embedding of \$\mathbb{R}^{n}\$ into the cotangent bundle \$T^{\ast }\mathbb{R}^{n}\$ of \$\mathbb{R}^{n}\$ that agrees with the cotangent fiber \$T_{x}^{\ast }\mathbb{R}^{n}\$ over a point \$x\neq 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 \$\mathbb{R}^{n}\setminus \{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^{\ast }\mathbb{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.