• DOI:10.2140/gt.2014.18.1115
• Corpus ID: 119338988

The period-index problem for twisted topological K-theory

@article{Antieau2014ThePP,
  title={The period-index problem for twisted topological K-theory},
  author={Benjamin Antieau and Ben Williams},
  journal={Geometry \& Topology},
  year={2014},
  volume={18},
  pages={1115-1148}
}

• Benjamin Antieau, B. Williams
• Published 24 April 2011
• Mathematics
• Geometry & Topology

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.