# Foliations on non-metrisable manifolds: absorption by a Cantor black hole - Mathematics > General Topology

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Abstract: We investigate contrasting behaviours emerging when studying foliations onnon-metrisable manifolds. It is shown that Kneser-s pathology of a manifoldfoliated by a single leaf cannot occur with foliations of dimension-one. On theother hand, there are open surfaces admitting no foliations. This is derivedfrom a qualitative study of foliations defined on the long tube $\mathbbS^1\times {\mathbb L} +$ product of the circle with the long ray, which isreminiscent of a `black hole-, in as much as the leaves of such a foliation arestrongly inclined to fall into the hole in a purely vertical way. Moregenerally the same qualitative behaviour occurs for dimension-one foliations on$M \times {\mathbb L} +$, provided that the manifold $M$ is -sufficientlysmall-, a technical condition satisfied by all metrisable manifolds. We alsoanalyse the structure of foliations on the other of the two simplest long pipesof Nyikos, the punctured long plane. We are able to conclude that the longplane $\mathbb L^2$ has only two foliations up to homeomorphism and six up toisotopy.

Autor: Mathieu Baillif, Alexandre Gabard, David Gauld

Fuente: https://arxiv.org/