Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Abstract

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 some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines R×t, t∊(0,1), and boundary intervals which gives a foliation Δ on all of Z. Denote by H(Z,Δ) the group of all homeomorphisms of Z that maps leaves of Δ onto leaves and by H(Z/Δ) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group π0H(Z,Δ) with a group of automorphisms of a certain graph G with additional structure which encodes the combinatorics of gluing Z from strips. That graph is in a certain sense dual to the space of leaves Z/Δ. On the other hand, for every h\inH(Z,Δ) the induced permutation k of leaves of Δ is in fact a homeomorphism of Z/Δ and the correspondence h→k is a homomorphism ψ:H(Δ)→H(Z/Δ). The aim of the present paper is to show that ψ induces a homomorphism of the corresponding homeotopy groups ψ0:π0H(Z,Δ)→π0H(Z/Δ) which turns out to be either injective or having a kernel Z2. This gives a dual description of π0H(Z,Δ) in terms of the space of leaves.