Exotic Nonleaves 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", that is, homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold, being the 1st examples in this class. [ABSTRACT FROM AUTHOR]