  The Topological Period-Index Conjecture is an 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$^c$ 6-manifolds. We also show that it fails in
general for 6-manifolds.
