Foliations with non-compact leaves on surfaces

TL;DR

This paper investigates foliations on non-compact surfaces formed by gluing strips, proving that the identity component of their homeomorphism group is contractible, thus revealing topological properties of these foliations.

Contribution

It introduces a new class of foliated non-compact surfaces constructed from strips and establishes the contractibility of their homeomorphism group's identity component.

Findings

The constructed surfaces admit foliations by parallel lines.

The identity path component of the homeomorphism group is contractible.

Provides insights into the topological structure of foliations on non-compact surfaces.

Abstract

We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.