Foliations on non-metrisable manifolds: absorption by a Cantor black hole

Abstract

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.