Symplectic Structures on the cotangent bundles of open 4-manifolds

Abstract

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^{\ast }{X}_0,T^{\ast }{X}_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to ${\mathbb{R}}^4$, the standard Stein structure on $T^{\ast }R$ is Stein homotopic to the standard Stein structure on $T^{\ast }{\mathbb{R}}^4={\mathbb{R}}^8$. We use this to show that any exotic ${\mathbb{R}}^4$ embeds in the standard symplectic ${\mathbb{R}}^8$ as a Lagrangian submanifold. As a corollary, we show that ${\mathbb{R}}^8$ has uncountably many smoothly distinct foliations by Lagrangian ${\mathbb{R}}^4$s with their standard smooth structure.