Basic Forms and Orbit Spaces: a Diffeological Approach

TL;DR

This paper extends the classical isomorphism between differential forms on quotient manifolds and basic forms to actions that are not necessarily free or proper, using a diffeological framework.

Contribution

It generalizes the isomorphism of differential forms to non-free, non-proper group actions via diffeological spaces, broadening the scope of previous results.

Findings

The isomorphism holds when the identity component acts properly.

Diffeological differential forms are used to handle non-manifold quotient spaces.

The classical result is extended beyond free and proper actions.

Abstract

If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense.