  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 some pairs of their boundary intervals. Every such strip has a
natural foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and
boundary intervals which gives a foliation $\Delta$ on all of $Z$. Denote by
$\mathcal{H}(Z,\Delta)$ the group of all homeomorphisms of $Z$ that maps leaves
of $\Delta$ onto leaves and by $\mathcal{H}(Z/\Delta)$ the group of
homeomorphisms of the space of leaves endowed with the corresponding compact
open topologies. Recently, the authors identified the homeotopy group
$\pi_0\mathcal{H}(Z,\Delta)$ with a group of automorphisms of a certain graph
$G$ with the additional structure which encodes the combinatorics of gluing $Z$
from strips. That graph is in a certain sense dual to the space of leaves
$Z/\Delta$.
  On the other hand, for every $h\in\mathcal{H}(Z,\Delta)$ the induced
permutation $k$ of leaves of $\Delta$ is in fact a homeomorphism of $Z/\Delta$
and the correspondence $h\mapsto k$ is a homomorphism
$\psi:\mathcal{H}(\Delta)\to\mathcal{H}(Z/\Delta)$. The aim of the present
paper is to show that $\psi$ induces a homomorphism of the corresponding
homeotopy groups
$\psi_0:\pi_0\mathcal{H}(Z,\Delta)\to\pi_0\mathcal{H}(Z/\Delta)$ which turns
out to be either injective or having a kernel $\mathbb{Z}_2$. This gives a dual
description of $\pi_0\mathcal{H}(Z,\Delta)$ in terms of the space of leaves.
