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 ∆ on all of Z. Denote by H(Z, ∆) the group of all homeomorphisms of Z that maps leaves of ∆ onto leaves and by H(Z/∆) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group π 0 H(Z, ∆) 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/∆.On the other hand, for every h ∈ H(Z, ∆) the induced permutation k of leaves of ∆ is in fact a homeomorphism of Z/∆ and the correspondence h → k is a homomorphism ψ : H(∆) → H(Z/∆). The aim of the present paper is to show that ψ induces a homomorphism of the corresponding homeotopy groups ψ 0 : π 0 H(Z, ∆) → π 0 H(Z/∆) which turns out to be either injective or having a kernel Z 2 . This gives a dual description of π 0 H(Z, ∆) in terms of the space of leaves.