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.

中文翻译：

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具有无限多端的奇异非叶

我们证明任何简单连通的拓扑封闭的$4$-流形沿着任何紧凑的、完全断开的驯服子集$\Lambda$ 都允许平滑的连续统，这与$C^{1,0}$ 余维的任何叶子都不是微分同胚的紧凑歧管上的一个叶面。这包括 $S^4$ 沿着驯服的康托集刺穿的非凡案例。这是这个实现问题的最低合理规律。这些结果来自 $C^{1,0}$ 规则中的非叶子的新标准。我们还在 $C^2$ 类别中包含了一个新的非叶子标准。我们的一些光滑非叶子是“奇异的”，也就是说，同胚但不微同胚于紧致流形上的同维叶一个叶，是此类中的第一个示例。