  We study the Postnikov tower of the classifying space of a compact Lie group
P(n,mn), which gives obstructions to lifting a topological Brauer class of
period $n$ to a PU_{mn}-torsor, where the base space is a CW complex of
dimension 8. Combined with the study of a twisted version of Atiyah-Hirzebruch
spectral sequence, this solves the topological period-index problem for CW
complexes of dimension 8.
