The Orbit Space and Basic Forms of a Proper Lie Groupoid

TL;DR

This paper generalizes the classical de Rham isomorphism from free proper Lie group actions to proper Lie groupoids, establishing a de Rham theorem for the orbit space with quotient diffeological structure.

Contribution

It extends the classical de Rham isomorphism to proper Lie groupoids and introduces a de Rham theorem for their orbit spaces with quotient diffeological structure.

Findings

Established an isomorphism between the de Rham complex of the orbit space and basic forms on the groupoid.

Proved a de Rham theorem for the orbit space of a proper Lie groupoid.

Generalized classical results to a broader class of geometric structures.

Abstract

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space.