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 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.

To appear in J. Top

References in corpus (2)

• Approximating classifying spaces by smooth projective varieties
• Surjectivity of the total Clifford invariant and Brauer dimension

Cited by in corpus (19)

• On the Cohomology of the Classifying Spaces of Projective Unitary Groups
• Gauging Lie group symmetry in (2+1)d topological phases
• The Topological Period-Index Problem over 8-Complexes, II
• The Topological Period-Index Problem over 8-Complexes, I
• The Topological Period-Index Conjecture for spin${}^c$ 6-manifolds
• Surjectivity of the total Clifford invariant and Brauer dimension
• Differential characters and cohomology of the moduli of flat Connections
• Involutions of Azumaya algebras
• Some torsion classes in the Chow ring and cohomology of $BPG{L}_n$
• The cohomology of projective unitary groups
• On The Topological Period-Index Problem over 8-manifolds
• On $H^{\ast }(BP{U}_n;\mathbb{Z})$ and Weyl group invariants
• The topological period-index conjecture
• On the Brown-Peterson cohomology of $BP{U}_n$ in lower dimensions and the Thom map
• Further counterexamples to the integral Hodge conjecture
• The cohomology of $BPU(p^m)$ and invariant polynomials
• A distinguished subring of the Chow ring and cohomology of $BPG{L}_n$
• Brauer class over the Picard scheme of totally degenerate stable curves
• Milnor operations and classifying spaces