## 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(ℤ/l,2), where l is a prime, we construct a sequence of spaces with an order l class in the Brauer group, but whose indices tend to infinity.

Original language | English (US) |
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Pages (from-to) | 1115-1148 |

Number of pages | 34 |

Journal | Geometry and Topology |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Apr 7 2014 |

Externally published | Yes |

## Keywords

- Brauer groups
- Cohomology of projective unitary groups
- Stable homotopy theory
- Twisted K(ℤ/l,2)-theory
- Twisted sheaves

## ASJC Scopus subject areas

- Geometry and Topology