The period-index problem for twisted topological K-theory

Bnjamin Antieau, Ben Williams

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

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 languageEnglish (US)
Pages (from-to)1115-1148
Number of pages34
JournalGeometry and Topology
Volume18
Issue number2
DOIs
StatePublished - Apr 7 2014
Externally publishedYes

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

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