The period-index problem for twisted topological K-theory

TL;DR

This paper addresses the period-index problem in twisted topological K-theory, providing bounds on the index of Brauer classes based on space dimension and class order, with applications to complex spaces and cohomological obstructions.

Contribution

It introduces a new approach to the period-index problem using twisted topological K-theory and cohomology of projective unitary groups, establishing bounds and obstructions.

Findings

Upper bounds on the index depending on space dimension and class order.

Construction of spaces with Brauer class of fixed order but unbounded index.

Cohomological obstructions to representing classes by Azumaya algebras.

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.