Exotic open $4$-manifolds which are non-leaves

TL;DR

This paper demonstrates the existence of uncountably many exotic open 4-manifolds, including some exotic R^4's and cylinders, that cannot be realized as leaves in certain compact foliations, revealing new insights into 4-manifold topology.

Contribution

It establishes the existence of uncountably many exotic open 4-manifolds that are not diffeomorphic to any leaf of a codimension one transversely C^2 foliation on a compact manifold.

Findings

Uncountably many exotic 4-manifolds cannot be realized as leaves.

Includes exotic R^4's and cylinders S^3×R.

Advances understanding of 4-manifold foliation properties.

Abstract

We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one transversely $C^{2}$ foliation on a compact manifold. These examples include some exotic ${\mathbb R}^4$'s and exotic cylinders $S^3\times{\mathbb R}$.