The Topological Period-Index Problem over 8-Complexes, I

TL;DR

This paper addresses the topological period-index problem for 8-dimensional CW complexes by analyzing the Postnikov tower of certain classifying spaces and employing a twisted spectral sequence to determine obstructions to lifting Brauer classes.

Contribution

It provides a solution to the topological period-index problem for 8-complexes using new methods involving Postnikov towers and twisted spectral sequences.

Findings

Solved the period-index problem for 8-dimensional CW complexes.

Identified obstructions to lifting Brauer classes via Postnikov towers.

Developed a twisted Atiyah-Hirzebruch spectral sequence approach.

Abstract

We study the Postnikov tower of the classifying space of a compact Lie group P(n,mn), which gives obstructions to lifting a topological Brauer class of period $n$ to a PU_{mn}-torsor, where the base space is a CW complex of dimension 8. Combined with the study of a twisted version of Atiyah-Hirzebruch spectral sequence, this solves the topological period-index problem for CW complexes of dimension 8.