Homeotopy groups of oneнdimensional foliations on surfaces.

Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R × (0, 1) with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines R × t, t 2∈ (0, 1), and boundary intervals, where we get a foliation Δ on all of Z. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "stripedِ" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane R2 by level-set of pseudo-harmonic functions R2 → R without singularities. Previously, the first two authors studied the homotopy type of the group H(Δ) of homeomorphisms of Z sending leaves of Δ into leaves, and shown that except for two cases the identity path component H0 (Δ) of H(Δ) is contractible. The aim of the present paper is to show that the quotient H(Δ)/H0(Δ) can be identied with the group of automorphisms of a certain graph with additional structure encoding the ُ combinatorics"ِ of gluing. [ABSTRACT FROM AUTHOR]