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
,
we give upper bounds on the index depending only on
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
–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
.
Applying this to the finite skeleta of the Eilenberg–Mac Lane space
,
where
is a prime, we construct a sequence of spaces with an order
class
in the Brauer group, but whose indices tend to infinity.
Keywords
Brauer groups, twisted $K\!$–theory, twisted sheaves,
stable homotopy theory, cohomology of projective unitary
groups