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

TL;DR

This paper proves the Topological Period-Index Conjecture for spin$^c$ 6-manifolds, establishing its validity and sharpness in this case, while also demonstrating its failure for general 6-manifolds.

Contribution

It confirms the conjecture for spin$^c$ 6-manifolds and clarifies its limitations for broader classes of 6-manifolds.

Findings

Conjecture holds for spin$^c$ 6-manifolds

Conjecture is sharp in this setting

Fails for some non-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.