The Topological Period-Index Conjecture is a hypothesis which relates the period
and index of elements of the cohomological Brauer group of a space. It was identified
by Antieau and Williams as a topological analogue of the Period-Index Conjecture
for function fields.
In this paper we show that the Topological Period-Index Conjecture holds and is in general
sharp for spin
-manifolds. We also show that
it fails in general for
-manifolds.