Homeotopy groups of one-dimensional foliations on surfaces

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 $mathbb{R}times t$, $tin(0,1)$, 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.