Exotic non-leaves with infinitely many ends

TL;DR

This paper demonstrates the existence of exotic smoothings of certain 4-manifolds that cannot be realized as leaves in codimension one foliations, introducing new criteria for nonleaves in various regularity classes.

Contribution

It introduces new criteria for identifying nonleaves in $C^{1,0}$ and $C^2$ regularity, and constructs exotic smoothings of 4-manifolds that are not diffeomorphic to leaves.

Findings

Existence of nonleaves in $C^{1,0}$ regularity.

Existence of nonleaves in the $C^2$ category.

Construction of exotic smoothings homeomorphic but not diffeomorphic to leaves.

Abstract

We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $\Lambda$ admits a continuum of smoothings which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are "exotic", i.e., homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold.