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The period-index problem for twisted topological K-theory

By Benjamin Antieau and Ben Williams

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.Comment: To appear in Geometry & Topology; minor cosmetic change

Topics: Mathematics - K-Theory and Homology, Mathematics - Algebraic Topology, 19L50, 16K50, 57T10, 55Q10, 55S35
Year: 2013
DOI identifier: 10.2140/gt.2014.18.1115
OAI identifier: oai:arXiv.org:1104.4654
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