Topology and purity for torsors

We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for $PGL_p$-torsors on regular noetherian schemes when $p$ is a prime. Extending our previous work when $p=2$, we obtain a negative answer to a question of Colliot-Th\'el\`ene and Sansuc, for all $PGL_p$. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.