Weak approximation for homogeneous spaces: reduction to the case with finite stabilizer

We reduce the question about whether the Brauer-Manin obstruction to weak approximation for homogeneous spaces is the only obstruction to the “simpler” question of the particular case of homogeneous spaces of ${\mathrm{S}\mathrm{L}}_n$ with finite stabilizer. We also give unconditional results based on the few known results on finite stabilizers.