Foliations with all nonclosed leaves on noncompact surfaces
Abstract
Let $X$ be a connected noncompact $2$dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a nonunique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1606.00045
 Bibcode:
 2016arXiv160600045M
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry;
 Mathematics  Dynamical Systems;
 Mathematics  General Topology;
 57R30;
 55R10
 EPrint:
 Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=884