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arXiv · 1208.4430

The topological period-index problem over 6-complexes

Benjamin Antieau·Ben Williams
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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 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.

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Benjamin Antieau, Ben Williams. 2012-08-22. The topological period-index problem over 6-complexes. https://doi.org/10.1112/jtopol%2Fjtt042

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