3-cocycle for a group action

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This term is defined, and makes sense, in the context of a group action on an Abelian group. In particular, it thus makes sense for a linear representation of a group

Definition

Let $G$ be a group acting (on the left) on an abelian group $A$ via a homomorphism of groups $\varphi :G\to \operatorname {Aut} (A)$ where $\operatorname {Aut} (A)$ is the automorphism group of $A$.

Explicit definition

A 3-cocycle for the action is a function $f:G\times G\times G\to A$ satisfying the following for all $g_{1},g_{2},g_{3},g_{4}\in G$ (here, the $g_{i}$s are allowed to be equal):

$\!\varphi (g_{1})(f(g_{2},g_{3},g_{4}))-f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})-f(g_{1},g_{2},g_{3}g_{4})+f(g_{1},g_{2},g_{3})=0$

If we suppress $\varphi$ and use $\cdot$ for the action, we can rewrite this as:

$\!g_{1}\cdot f(g_{2},g_{3},g_{4})-f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})-f(g_{1},g_{2},g_{3}g_{4})+f(g_{1},g_{2},g_{3})=0$

or equivalently:

$\!g_{1}\cdot f(g_{2},g_{3},g_{4})+f(g_{1},g_{2}g_{3},g_{4})+f(g_{1},g_{2},g_{3})=f(g_{1}g_{2},g_{3},g_{4})+f(g_{1},g_{2},g_{3}g_{4})$

Note that a function $f:G\times G\times \to A$ (without any conditions) is sometimes termed a 3-cochain for the group action.

Definition as part of the general definition of cocycle

A 3-cocycle for a group action is a special case of a cocycle for a group action, namely $n=3$. This, in turn, is the notion of cocycle corresponding to the Hom complex from the bar resolution of $G$ to $A$ as $\mathbb {Z} G$-modules.

Group structure

The set of 3-cocycles for the action of $G$ on $A$ forms a group under pointwise addition.

As a group of homomorphisms

For any group $G$, we can construct a $\mathbb {Z} G$-module $K$ such that for any abelian group $A$, the group of 3-cocycles $f:G\times G\to A$ can be identified with the group of $\mathbb {Z} G$-module maps from $K$ to $A$.

Importance

• See 3-cocycle for trivial group action#Importance