Exotic open $4$–manifolds which are nonleaves

Abstract

We study the possibility of realizing exotic smooth structures on finitely punctured simply connected closed $4$–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 $C^2$ foliation on a compact manifold. These examples include some exotic ${ℝ}^4$’s and exotic cylinders $S^3\times ℝ$.