  We prove that for every reductive algebraic group $H$ with centre of positive
dimension and every integer $K$ there is a smooth and projective variety $X$
and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to
\Bclass H$ induces an isomorphism in cohomology in degrees $\le K$. This is
then applied to show that if $G$ is a connected non-special group there is a
$G$-torsor $P \to X$ for which we do not have $[P]=[G][X]$ in the (completion
of the) Grothendieck ring of varieties.
