The period-index problem for twisted topological K–theory

Antieau, Benjamin¹ ; Williams, Ben²

¹ Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA
² Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

Geometry & topology, Tome 18 (2014) no. 2, pp. 1115-1148.

Résumé

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 $K\left(\mathbb{Z}∕\ell ,2\right)$, where $\ell$ is a prime, we construct a sequence of spaces with an order $\ell$ class in the Brauer group, but whose indices tend to infinity.

Détail

DOI : 10.2140/gt.2014.18.1115

Classification : 16K50, 19L50, 55S35
Keywords: Brauer groups, twisted $K\!$–theory, twisted sheaves, stable homotopy theory, cohomology of projective unitary groups

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