Baillif, MGabard, AGauld, David2012-06-262009arXiv:0910.1897https://hdl.handle.net/2292/19173We 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.Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmJournal ArticleCopyright: the Authorhttp://purl.org/eprint/accessRights/RestrictedAccess