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-Thèléne 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.