Generic freeness in frame bundle prolongations of $C^\infty$ actions

TL;DR

This paper proves that for smooth actions of real Lie groups on manifolds with certain fixpoint properties, there exists a higher order frame bundle where the group acts freely outside a meager invariant set.

Contribution

It establishes a generic freeness result for higher order frame bundle prolongations of smooth Lie group actions with nowhere dense fixpoint sets.

Findings

Existence of a G-invariant meager subset where the action is free

Applicable to submanifold jet bundles

Extends freeness results to higher order prolongations

Abstract

Let a real Lie group $G$ act on a $C^\infty$ real manifold $M$. Assume that the action is $C^\infty$ and that every nontrivial element of $G$ has a nowhere dense fixpoint set in $M$. We show that, in~some higher order frame bundle $F$ of $M$, there exists a $G$-invariant meager subset $Z$ of $F$ such that the $G$-action on $F\backslash Z$ is free. A similar result holds for submanifold jet bundles.