Unramified Brauer groups of finite and infinite groups

TL;DR

This paper introduces a homological approach to the Bogomolov multiplier, linking it to unramified Brauer groups, and provides new formulas, descriptions, and an algorithm for its computation, with implications for group theory and K-theory.

Contribution

It develops a homological framework for the Bogomolov multiplier, including a Hopf-type formula, exact sequence, and a new description for nilpotent groups, also connecting it to K-theory.

Findings

Derived a homological version of the Bogomolov multiplier

Proved a Hopf-type formula and a five-term exact sequence

Developed an algorithm for computing the Bogomolov multiplier

Abstract

The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a Hopf-type formula, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. A new description of the Bogomolov multiplier of a nilpotent group of class two is obtained. We define the Bogomolov multiplier within K-theory and show that proving its triviality is equivalent to solving a long-standing problem posed by Bass. An algorithm for computing the Bogomolov multiplier is developed.