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Unramified Brauer groups of finite and infinite groups. (English) Zbl 1346.20072

Summary: 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.

MSC:

20J05 Homological methods in group theory
14F22 Brauer groups of schemes
19C09 Central extensions and Schur multipliers

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