Homeotopy groups of one-dimensional foliations on surfaces

TL;DR

This paper studies the structure of homeomorphism groups of certain one-dimensional foliations on non-compact surfaces, revealing their relation to automorphism groups of associated combinatorial graphs.

Contribution

It characterizes the quotient of the homeomorphism group by its identity component as an automorphism group of a structured graph, extending previous homotopy results.

Findings

The identity component of the homeomorphism group is contractible in most cases.

The quotient group corresponds to automorphisms of a combinatorial graph.

Provides a topological classification of leaf-preserving homeomorphisms.

Abstract

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $\Delta$ on all of $Z$. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "striped" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane $\mathbb{R}^2$ by level-set of pseudo-harmonic functions $\mathbb{R}^2 \to \mathbb{R}$ without singularities. Previously, the first two authors studied the homotopy type of the group $\mathcal{H}(\Delta)$ of homeomorphisms of $Z$ sending leaves of $\Delta$ onto leaves, and shown that except for two cases the identity path component $\mathcal{H}_{0}(\Delta)$ of $\mathcal{H}(\Delta)$ is contractible. The aim of the present paper is to show that the quotient $\mathcal{H}(\Delta)/ \mathcal{H}_{0}(\Delta)$ can be identified with the group of automorphisms of a certain graph with additional structure encoding the "combinatorics" of gluing.