The Topological Period-Index Conjecture for spin$^c$ 6-manifolds

Abstract: 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.