The Topological Period-Index Problem over 8-Complexes, I

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.