# 3-cocycles on outer automorphism groups

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.

• 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. Apr 21, 2022 at 8:21
• @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. Apr 22, 2022 at 11:46

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

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