Zéro-cycles de degré un sur les espaces homogènes

We explain how to build homogeneous spaces of a connected linear algebraic group, having zero-cycles of degree one but no rational point, even on a smooth compactification. Thus, we give a negative answer to a recent question asked by Burt Totaro. Parimala ([Pa]) has independently produced such spaces over fields of the type k((t)) (for k a suitable p-adic field), which are projective varieties. We use here a totally different approach: we apply a nonabelian cohomological machinery to a given extension of finite groups which is non-split, but split over every p-Sylow of the base. We give two precise examples: the first and easiest one, over the field C((x))((y)); the second one, which requires a little more work, over a local or global field, and with finite abelian stabilizers. Both are rational varieties. 1