Symplectic Structures on the cotangent bundles of open 4-manifolds

TL;DR

This paper proves that cotangent bundles of homeomorphic open 4-manifolds are symplectomorphic, and explores implications for exotic ers, Stein structures, and Lagrangian embeddings in symplectic geometry.

Contribution

It establishes symplectomorphism between cotangent bundles of homeomorphic open 4-manifolds and investigates Lagrangian embeddings of exotic ers.

Findings

Cotangent bundles of homeomorphic open 4-manifolds are symplectomorphic.

Exotic ers embed as Lagrangian submanifolds in standard er.

Uncountably many distinct foliations of er by Lagrangian ers.

Abstract

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. 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.