Actions of groups of foliated homeomorphisms on spaces of leaves

Abstract

Let $\Delta$ be a foliation on a topological manifold $X$, $Y$ be the space of leaves, and $p: X \to Y$ be the natural projection. Endow $Y$ with the factor topology with respect to $p$. Then the group $\mathcal{H}(X, \Delta)$ of foliated (i.e. mapping leaves onto leaves) homeomorphisms of $X$ naturally acts on the space of leaves $Y$, which gives a homomorphism $\psi: \mathcal{H}(X, \Delta) \to \mathcal{H}(Y)$. We present sufficient conditions when $\psi$ is continuous with respect to the corresponding compact open topologies. In fact similar results hold not only for foliations but for a more general class of partitions $\Delta$ of locally compact Hausdorff spaces $X$.