Exotic non-leaves with infinitely many ends

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.