Actions of groups of foliated homeomorphisms on spaces of leaves

TL;DR

This paper investigates how groups of leaf-preserving homeomorphisms of foliated manifolds act on the space of leaves, providing conditions for the continuity of this action in the context of topological foliations and partitions.

Contribution

It establishes sufficient conditions for the continuity of the induced action of foliated homeomorphism groups on leaf spaces, extending results to general partitions of locally compact Hausdorff spaces.

Findings

Conditions for continuity of the action homomorphism

Extension of results from foliations to general partitions

Applicability to locally compact Hausdorff spaces

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 $X$.