  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 at most 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.
