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–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.