Basic Forms and Orbit Spaces: a Diffeological Approach

Abstract

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.

The Basic de Rham Complex of a Singular Foliation

“…A key feature of the above arguments is that linearizability of a (source-connected) Lie groupoid suffices to guarantee it has property (P). Since properness is sufficient but not necessary for linearizability, our class of examples for which (P) holds is distinct from those found in [10] and [21]. However, even linearizability is not a necessary condition for (P).…”

“…3, we show that this implies the singular foliation of M by the G • orbits satisfies the hypothesis in (B), hence has property (P). The fact G • is connected implies G • ⋉ M ⇒ M also has property (P) (Proposition 5.6), and a lemma from [10] shows this suffices to conclude G ⋉ M ⇒ M has property (P).…”

“…Item (B) has not appeared in the literature, and extends a few existing results. In [10], Karshon and Watts proved that given a Lie group G acting on M whose identity component G • acts properly, the corresponding action groupoid G ⋉ M ⇒ M has property (P). We can alternatively get this as a consequence of (B).…”

“…In [21], Watts extended [10] to prove that proper Lie groupoids have property (P). Assuming source-connectedness, we give another argument.…”

Correction: Bates et al. Vector Fields and Differential Forms on the Orbit Space of a Proper Action. Axioms 2021, 10, 118

“…The authors would like to thank Editor Luna Shen for her invitation to publish a feature paper in Axioms. The authors are grateful to Gerald Schwarz for pointing out an error in the proof of Lemma 5, and for suggesting the inclusion in the bibliography of three additional papers [2][3][4], related to the problem under consideration.…”