Differential Forms on Orbit Spaces

Differential forms are mathematical constructs used in a wide variety of contexts: they are what you integrate in calculus, they provide extra structure to spaces such as curvature or a symplectic structure, and these in turn are used to describe physical systems. Moreover, differential forms are important for obtaining topological invariants, such as de Rham cohomology. In this talk, we will discuss differential forms in the context of symmetry: a famous result of Koszul states that the singular cohomology of an orbit space coming from a compact Lie group action is the same as the cohomology of the basic differential forms (forms that respect the symmetry). In fact, if the action is free (trivial stabilisers), then the de Rham complex of forms on the orbit space is isomorphic to the complex of basic differential forms. Using the theory of diffeology, this relationship between differential forms extends to the non-free case; that is, we allow the orbit space to have singularities. Consequently, we have a triple of isomorphic cohomology theories: that from basic differential forms, singular cohomology on the orbit space, and diffeological de Rham cohomology on the orbit space.