The topological period-index problem over 6-complexes

TL;DR

This paper solves the topological period-index problem for 6-dimensional complexes by analyzing spectral sequences and Postnikov towers, providing new obstructions and examples related to Brauer classes and Clifford algebras.

Contribution

It establishes a full solution to the topological period-index problem for complexes up to dimension 6 and introduces conditions that challenge existing conjectures in algebraic geometry.

Findings

Lower bound on topological index in terms of period

Obstructions to realizing certain Brauer classes as Clifford algebra classes

Construction of varieties where the total Clifford invariant map is not surjective

Abstract

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension at most 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.