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 PUmn-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.

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