The topological period-index problem over 6-complexes

Abstract

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah–Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period–index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period–index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.