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):

$\!\varphi(g_1)(f(g_2,g_3,g_4)) - f(g_1g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) - f(g_1,g_2,g_3g_4) + f(g_1,g_2,g_3) = 0$

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

$\!g_1 \cdot f(g_2,g_3,g_4) - f(g_1g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) - f(g_1,g_2,g_3g_4) + f(g_1,g_2,g_3) = 0$

or equivalently:

$\! g_1 \cdot f(g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) + f(g_1,g_2,g_3) = f(g_1g_2,g_3,g_4) + f(g_1,g_2,g_3g_4)$

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