Let be a compact Lie group and let be a smooth -manifold. If happens to act locally freely on , then the quotient of by the -action is an example of an orbifold. In the study of the geometry of orbifolds, an object called the inertia orbifold has played a major role. The inertia orbifold is a disjoint union of orbifolds given by the quotient of the space of loops of the translation groupoid, a smooth manifold, by a natural action of the translation groupoid itself.

If the action of is not assumed to be locally free, then the space of loops of the translation groupoid is no longer a smooth manifold and the quotient is no longer an orbifold. In this case, we refer to the quotient of the space of loops as the inertia space of the -manifold . We will describe an explicit Whitney stratification of the inertia space. Using this stratification, we will present a de Rham theorem for cohomology of differential forms on the inertia space with respect to this stratification. (Joint work with Carla Farsi and Markus Pflaum)