Paper detail

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

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 $Δ$ on all of $Z$. Denote by $\mathcal{H}(Z,Δ)$ the group of all homeomorphisms of $Z$ that maps leaves of $Δ$ onto leaves and by $\mathcal{H}(Z/Δ)$ the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group $π_0\mathcal{H}(Z,Δ)$ 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 $Z/Δ$. On the other hand, for every $h\in\mathcal{H}(Z,Δ)$ the induced permutation $k$ of leaves of $Δ$ is in fact a homeomorphism of $Z/Δ$ and the correspondence $h\mapsto k$ is a homomorphism $ψ:\mathcal{H}(Δ)\to\mathcal{H}(Z/Δ)$. The aim of the present paper is to show that $ψ$ induces a homomorphism of the corresponding homeotopy groups $ψ_0:π_0\mathcal{H}(Z,Δ)\toπ_0\mathcal{H}(Z/Δ)$ which turns out to be either injective or having a kernel $\mathbb{Z}_2$. This gives a dual description of $π_0\mathcal{H}(Z,Δ)$ in terms of the space of leaves.