Symplectic Structures on the Cotangent Bundles of Open 4-Manifolds

We show that, for any two orientable smooth open 4-manifolds X-0 and X-1, which are homeomorphic, their cotangent bundles T*X-0 and T*X-1 are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold R homeomorphic to R-4, the standard Stein structure on T* R is Stein homotopic to the standard Stein structure on T*R-4 = R-8. We use this to show that any exotic R-4 embeds in the standard symplectic R-8 as a Lagrangian submanifold. As a corollary, we show that R-8 has uncountably many smoothly distinct foliations by Lagrangian R(4)s with their standard smooth structure.