Abstract
Let Z be a non-compact two-dimensional manifold obtained from a family of open strips RĂ(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 RĂt, tâ(0,1), and boundary intervals which gives a foliation Î on all of Z. Denote by H(Z,Î) the group of all homeomorphisms of Z that maps leaves of Î onto leaves and by H(Z/Î) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group Ï0H(Z,Î) 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/Î.
 On the other hand, for every h\inH(Z,Î) the induced permutation k of leaves of Î is in fact a homeomorphism of Z/Î and the correspondence hâk is a homomorphism Ï:H(Î)âH(Z/Î). The aim of the present paper is to show that Ï induces a homomorphism of the corresponding homeotopy groups Ï0:Ï0H(Z,Î)âÏ0H(Z/Î) which turns out to be either injective or having a kernel Z2. This gives a dual description of Ï0H(Z,Î) in terms of the space of leaves.
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