Given a group $G$, the outer automorphism group $Out(G)$ acts on the center by $Z(G)$ by lifting an outer automorphism to an actual automorphism and evaluating this on elements of $Z(G)$. What is classified by the degree three group cohomology $H^3(Out(G),Z(G))$ ?

Given an algebra $A$, say finite-dimensional over a field, the outer automorphism group $Out(A)$ acts on the group of central units $Z(A)^\times$ by lifting an outer automorphism to an actual automorphism and then evaluating. The action is trivial if $A$ is a central algebra. What is classified by the degree three group cohomology $H^3(Out(A),Z(A)^\times)$ ?

My motivation for this question is the following. If $A$ is my algebra, then we have a crossed module $$ A^\times \to Aut(A) $$ given by inner automorphisms and the evaluation action of $Aut(A)$ on $A^\times$. The homotopy groups of this crossed module are $\pi_0=Out(A)$ and $\pi_1=Z(A)^\times$, and the usual action of $\pi_0$ on $\pi_1$ is the one described above. Crossed modules have a so-called k-invariant, which is precisely a class $\xi\in H^3(\pi_0,\pi_1)$.

Baez and Lauda have shown that crossed modules are classified in a sense by triples $(\pi_0,\pi_1,\xi)$. The classification is saying that the crossed module - viewed as a monoidal category - is equivalent to the usual monoidal category constructed from the 3-cocycle $\xi$.

Baez, John C.; Lauda, Aaron D., Higher-dimensional algebra. V: 2-Groups, Theory Appl. Categ. 12, 423-491 (2004). ZBL1056.18002.

It is also known that if $A$ and $B$ are Picard-surjective and Morita equivalent, then their crossed modules are equivalent and so their classes coincide. Moreover, if $A$ is Picard-surjective and central-simple, then its class vanishes.

Summarizing, associated to every algebra is a class in $H^3(Out(A),Z(A)^\times)$. What is the intrinsic meaning of this class, apart from classifying some crossed module?

For groups instead of algebras it is basically the same story, and the question is analogous.

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    $\begingroup$ Closely related (duplicate?) mathoverflow.net/q/143031/41291 In particular (my) answer there mentions work of Hübschmann relating these 3-cocycles to class field theory, Dixmier-Douady classes and to the stuff by Jones, Rieffel and others related to crossed products of von Neumann algebras and such. $\endgroup$ Apr 21, 2022 at 8:21
  • $\begingroup$ @მამუკაჯიბლაძე: Sorry, I can't quite see any relation. My question is not about general 3-cocycles, but it is about 3-cocycles on outer automorphism groups. $\endgroup$ Apr 22, 2022 at 11:46

1 Answer 1


Among other things, the third cohomology contains an invariant for the existence of group-graded algebras whose degree-1-piece is $A$ / group extension with $G$ as normal subgroup. This is a theorem of Schreier. If I'm not mistaken, the cohomology class $\xi$ that comes from crossed modules is precisely Schreier's invariant.

There is a generalisation of both of these with more abstract nonsense sprinkled in (2-groups among them) see this earlier question of mine

  • $\begingroup$ This seems to be it, thank you! $\endgroup$ Apr 22, 2022 at 11:50
  • $\begingroup$ What is a good reference for Schreier's invariant? $\endgroup$ Apr 22, 2022 at 11:52
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    $\begingroup$ The list in the nlab article seems promising $\endgroup$ Apr 22, 2022 at 22:58

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