# Persistence of Freeness for Lie Pseudogroup Actions

## TL;DR

This paper proves that the local and global freeness of a Lie pseudogroup's action on jet spaces persists under prolongation, supporting the construction of complete moving frames and invariant analysis.

## Contribution

It establishes the persistence of freeness in Lie pseudogroup actions on jet spaces, foundational for invariant object construction.

## Key findings

- Freeness of pseudogroup actions persists under prolongation.
- Supports the construction of complete moving frames.
- Facilitates analysis of invariants in jet spaces.

## Abstract

The action of a Lie pseudogroup $G$ on a smooth manifold $M$ induces a prolonged pseudogroup action on the jet spaces $J^n$ of submanifolds of $M$. We prove in this paper that both the local and global freeness of the action of $G$ on $J^n$ persist under prolongation in the jet order $n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the pseudogroup action on $J^\infty$.