Abstract

For a large family of properties of homogeneous spaces, we prove that such a property holds for all homogeneous spaces of connected linear algebraic groups as soon as it holds for homogeneous spaces of SLn with finite stabilisers. As an example, we reduce to this particular case an important conjecture by Colliot-Thelene about the Brauer-Manin obstruction to the Hasse principle and to weak approximation. A recent work by Harpaz and Wittenberg proves that the main result also applies to the analogous conjecture (known as conjecture (E)) for zero-cycles on homogeneous spaces.