Foliations with all non-closed leaves on non-compact surfaces

Abstract

Let $X$ be a connected non-compact $2$-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 non-unique 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 $\Delta$.

Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=884

Cited by in corpus (4)

• Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold
• Smooth approximations and their applications to homotopy types
• Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves
• One-dimensional foliations on topological manifolds