Volume 18, issue 2 (2014)

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The period-index problem for twisted topological $K\!$–theory

Benjamin Antieau and Ben Williams

Geometry & Topology 18 (2014) 1115–1148

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–Mac Lane space K(,2), where is a prime, we construct a sequence of spaces with an order class in the Brauer group, but whose indices tend to infinity.

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Brauer groups, twisted $K\!$–theory, twisted sheaves, stable homotopy theory, cohomology of projective unitary groups
Mathematical Subject Classification 2010
Primary: 16K50, 19L50
Secondary: 55S35
Received: 24 April 2011
Accepted: 1 December 2013
Published: 7 April 2014
Proposed: Ralph Cohen
Seconded: Haynes Miller, Bill Dwyer
Benjamin Antieau
Department of Mathematics
University of Washington
Box 354350
Seattle, WA 98195
Ben Williams
Department of Mathematics
University of British Columbia
1984 Mathematics Road
Vancouver, BC V6T 1Z2