TITLE: The Teichmüller's algebraic interpretation of $H^3$ in group cohomology
QUESTION [17 upvotes]: In the book "Cohomology of Groups" of Kenneth S. Brown, it is told in the introduction that Teichmüller arrived to $H^3$ in an algebraic context, i.e. that Teichmüller worked with an algebraic interpretation of $H^3$ in group cohomology. I have tried to find the original article Über die sogenannte nichtcommutative Galoische Theorie und die Relation $\xi_{\lambda,\mu,\nu}\xi_{\lambda,\mu\nu,\pi}\xi^\lambda_{\mu,\nu,\pi}=\xi_{\lambda,\mu,\nu\pi}\xi_{\lambda\mu,\nu,\pi}$ of Teichmüller, but I haven't found it.
So here is my question, which is this algebraic interpretation of $H^3$ in group cohomology? Or, in another way, which was the algebraic context in which Teichmüller was working to arriving to $H^3$?

REPLY [3 votes]: In "Normality of algebras over commutative rings and the Teichmüller class. I" (II, III, arXiv version 1512.07264), Hübschmann constructs an explicit crossed module representing the Teichmüller cocycle and, among other things, reinterprets in terms of it the Deuring extension obstruction which seems to have been the original motivation of Teichmüller: very roughly, given an algebra $A$ with a group action, one seeks an embedding $A\subset C$ into a larger algebra which realizes the action via conjugation by (invertible) elements of $C$. You can find lots of references to previous work there, although, it seems, nobody before considered the corresponding crossed modules.
What I find especially interesting, in the third of these papers Hübschmann explains how this construction relates to Dixmier-Douady classes and work of Jones, Rieffel and others concerning group actions on C*-algebras, crossed products of von Neumann algebras and such.