Non-metrisable Manifolds pp 129-152 | Cite as

# Foliations on Non-metrisable Manifolds

## Abstract

Following Chap. 7 we present our second method of transferring a local structure from euclidean space to a manifold: foliations. In \({\mathbb R}^n\) we have an affine structure which leads to a partition of \({\mathbb R}^n\) into \(\mathfrak c\) many affine subspaces of some fixed dimension, for example parallel lines in \({\mathbb R}^n\) or parallel planes in \({\mathbb R}^3\). Transferring this to a manifold gives what is called a foliation of the manifold with the sets corresponding to the parallel lines, planes, etc. being called leaves. Just as we may partition \({\mathbb R}^n\) into sets of the form \({\mathbb R}^p\times \{y\}\), for \(y\in {\mathbb R}^{n-p}\), for each \(p=1,\dots , n-1\), so we may try to foliate a manifold into a collection of ‘parallel’ leaves of any (fixed) dimension from 1 to \(n-1\). Of course these leaves may well extend well beyond any particular coordinate chart. We present the definition and some examples in Sect. 8.1. In Sect. 8.2 we investigate dimension 1 foliations on certain ‘long’ manifolds, especially manifolds of the form \(M\times {\mathbb L}_+\), where \(M\) is a ‘small’ manifold, for example metrisable: we find that if there is at least one non-metrisable leaf then from some point \({\alpha }\) on the foliation of \(M\times {\mathbb L}_+\) is just the trivial foliation whose leaves are of the form \(\{x\}\times ({\alpha },{\omega }_1)\) for \(x\in M\). We then apply this in Sect. 8.3 to the long cylinder \({\mathbb S}^1\times {\mathbb L}_+\) where we find that the end behaves rather like a black hole with leaves either often circulating around with constant \({\mathbb L}_+\) coordinate or falling straight to the end with constant \({\mathbb S}^1\) coordinate. In the final section, Sect. 8.4, we discuss foliations of the long plane \({\mathbb L}^2\), initially with a compact subset removed. One surprise is that the long plane \({\mathbb L}^2\) supports very few distinct foliations, unlike the real plane \({\mathbb R}^2\).

## References

- 1.Baillif, M., Gabard, A., Gauld, D.: Foliations on non-metrisable manifolds: absorption by a Cantor black hole. Proc. Amer. Math. Soc.
**142**, 1057–1069 (2014)Google Scholar - 2.Baillif, M., Gabard, A., Gauld, D.: Foliations on non-metrisable manifolds: absorption by a Cantor black hole. (Expanded version of [1] found at arXiv, http://arxiv.org/abs/0910.1897)
- 3.Baillif, M., Gabard, A., Gauld, D.: Foliations on non-metrisable manifolds II: contrasted behaviours. arXiv, http://arxiv.org/abs/1303.6714
- 4.Ben Ami, E., Rubin, M.: On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds. Topol. Appl.
**157**, 1664–1679 (2010)Google Scholar - 5.Kneser, H.: Abzählbarkeit und geblätterte Mannigfaltigkeiten. Arch. Math.
**13**, 508–511 (1962)CrossRefzbMATHMathSciNetGoogle Scholar - 6.Kneser, M.: Beispiel einer dimensionserhöhenden analytischen Abbildung zwischen überabzählbaren Mannigfaltigkeiten. Arch. Math.
**11**, 280–281 (1960)CrossRefzbMATHGoogle Scholar - 7.Lawson, H.B.: Foliations. Bull. Amer. Math. Soc.
**80**, 369–418 (1974)Google Scholar - 8.Milnor, J.W.: Foliations and foliated vector bundles. Notes from lectures given at MIT, Fall 1969, http://www.foliations.org/surveys/FoliationLectNotes_Milnor.pdf (1969)
- 9.Nyikos, P.: The topological structure of the tangent and cotangent bundles on the long line. Topology Proceedings
**4**, 271–276 (1979)Google Scholar