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

AMSart, 15 pages, 1 figure

References in corpus (2)

• Smooth approximations and their applications to homotopy types
• Homeotopy groups of rooted tree like non-singular foliations on the plane