# On locally trivial extensions of topological spaces by a pseudogroup

## TL;DR

This paper investigates locally trivial extensions of topological spaces by a specific pseudogroup derived from Lie group actions, with applications to Riemannian foliations and their holonomy pseudogroups.

## Contribution

It generalizes the case of holonomy pseudogroups of Riemannian foliations to broader classes of locally trivial extensions by pseudogroups.

## Key findings

- Characterization of locally trivial extensions by a pseudogroup $\Gamma imes G$
- Application to the structure of holonomy pseudogroups in Riemannian foliations
- Extension of Molino's theory to more general pseudogroup actions

## Abstract

In this paper we restrict ourselves to the particular case where the pseudogroup is $\Gamma \ltimes G$ given by the action of a dense subgroup $\Gamma$ of a Lie group $G$ acting on $G$ by left translations. For a Riemannian foliation $F$ on a complete Riemannian manifold $M$ which is transversally parallelizable in the sense of Molino, let $X$ be the space of leaves closures. The holonomy pseudogroup of $F$ is an example of a locally trivial extension of $X$ by $\Gamma \ltimes G$. The study of a generalization of this particular case shall be the main purpose of this paper.