不可度量化流形上的叶线：Cantor黑洞的吸收
Foliations on non-metrisable manifolds: absorption by a Cantor black hole
摘要：
我们研究对比行为时出现的非度量流形上的叶理。它表明，Kneser的病理学的歧管叶状由一个单一的叶子不能发生叶状的维度1。另一方面，有开放的表面承认没有叶理。这是从定性研究的叶理上定义的长管$\mathbb S^1\times {\mathbb L}_+$（产品的圆与长射线），这是让人想起一个“黑洞”，尽可能多的叶片，这样的叶理是强烈倾向于落入孔在一个纯粹的垂直方式。更一般地说，同样的定性行为发生在$M \times {\mathbb L}_+$上的一维叶理上，只要流形$M$“足够小”，这是所有可度量流形都满足的技术条件。我们还分析了叶理的结构上的其他两个最简单的长管的Nyikos，穿孔长飞机。我们可以得出结论，长平面$\mathbb L^2$只有两个叶理直到同胚和六个直到合痕。
英文摘要：
We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. It is shown that Kneser's pathology of a manifold foliated by a single leaf cannot occur with foliations of dimension-one. On the other hand, there are open surfaces admitting no foliations. This is derived from a qualitative study of foliations defined on the long tube $\mathbb S^1\times {\mathbb L}_+$ (product of the circle with the long ray), which is reminiscent of a `black hole', in as much as the leaves of such a foliation are strongly inclined to fall into the hole in a purely vertical way. More generally the same qualitative behaviour occurs for dimension-one foliations on $M \times {\mathbb L}_+$, provided that the manifold $M$ is "sufficiently small", a technical condition satisfied by all metrisable manifolds. We also analyse the structure of foliations on the other of the two simplest long pipes of Nyikos, the punctured long plane. We are able to conclude that the long plane $\mathbb L^2$ has only two foliations up to homeomorphism and six up to isotopy.
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