The de Rham cohomology of a Lie group modulo a dense subgroup

Let $H$ be a dense subgroup of a Lie group $G$ with Lie algebra $\mathfrak g$. We show that the (diffeological) de Rham cohomology of $G/H$ equals the Lie algebra cohomology of $\mathfrak g/\mathfrak h$, where $\mathfrak h$ is the ideal ${Z\in\mathfrak g:\exp(tZ)\in H \text{ for all } t\in\mathbf R}$.

💡 Research Summary

In this paper Brant Clark and François Ziegler address the problem of computing de Rham cohomology for quotients G/H when H is a dense (hence non‑closed) subgroup of a Lie group G. The usual topological approach fails: the subset topology on H is not a Lie group topology, and the quotient topology on G/H is either non‑Hausdorff or even trivial when H is dense. To overcome this, the authors work in the category of diffeological spaces, introduced by Souriau, which admits arbitrary sub‑objects and quotients while still carrying a well‑behaved de Rham complex.

The key objects are:

• The Lie algebra 𝔤 of G and the ideal
  \