The period-index problem for twisted topological K-theory

Abstract

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension d, we give upper bounds on the index depending only on d and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological K-theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree n. Applying this to the finite skeleta of the Eilenberg-MacLane space K(Z/l,2), where l is a prime, we construct a sequence of spaces with an order l class in Br, but whose indices tend to infinity.

To appear in Geometry & Topology; minor cosmetic changes

References in corpus (2)

• Twisted sheaves and the period-index problem
• Cohomological obstruction theory for Brauer classes and the period-index problem

Cited by in corpus (9)

• The topological period-index problem over 6-complexes
• On the Cohomology of the Classifying Spaces of Projective Unitary Groups
• The Topological Period-Index Problem over 8-Complexes, I
• The Topological Period-Index Problem over 8-Complexes, II
• Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence
• The Topological Period-Index Conjecture for spin${}^c$ 6-manifolds
• Azumaya Algebras Without Involution
• Some torsion classes in the Chow ring and cohomology of $BPG{L}_n$
• The cohomology of $BPU(p^m)$ and invariant polynomials