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 $T<sup>*X_0,T<sup>*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}<sup>4$, the standard Stein structure on $T<sup>*R$ is Stein homotopic to the standard Stein structure on $$T^{*\mathbb{R}⁴} = \mathbb{R}<sup>8$$. We use this to show that any exotic $\mathbb{R}<sup>4$ embeds in the standard symplectic $\mathbb{R}<sup>8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}<sup>8$ has uncountably many smoothly distinct foliations by Lagrangian $\mathbb{R}<sup>4$s with their standard smooth structure.