Let H H be a dense subgroup of a Lie group G G with Lie algebra g {g}. We show that the (diffeological) de Rham cohomology of G / H G/H equals the Lie algebra cohomology of g / h {g}/ {h}, where h {h} is the ideal Z ∈ g: exp (t Z) ∈ H for all t ∈ R \Z { {g}: (tZ) H for all t R\}.
Clark et al. (Fri,) studied this question.