Cocycle for a group action

Definition

Suppose $G$ is a group and $A$ is an abelian group, with an action of $G$ on $A$.

Definition in terms of bar resolution

A $n$-cocycle is an element in the $n^{th}$ cocycle group for the Hom complex from the bar resolution of $G$ to $A$, in the sense of $\mathbb {Z} G$-modules.

Explicit definition

For $n$ a nonnegative integer, a $n$-cocycle for the action $\varphi$ of $G$ on $A$ is a function $f:G^{n}\to A$ such that, for all $g_{1},g_{2},\dots ,g_{n+1}\in G$:

$\!g_{1}\cdot f(g_{2},g_{3},\dots ,g_{n+1})+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]-f(g_{1},g_{2},\dots ,g_{n})=0$

In particular, when the action is trivial, this is equivalent to saying that:

$\!f(g_{2},g_{3},\dots ,g_{n+1})+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]-f(g_{1},g_{2},\dots ,g_{n})=0$

A cocycle for a group action

Particular cases

A 1-cocycle

Further information: 1-cocycle for a group action

A 1-cocycle is a function $f:G\to A$ such that:

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

In particular,a 1-cocycle for the trivial group action is a homomorphism of groups from $G$ to $A$.

A 2-cocycle

Further information: 2-cocycle for a group action, 2-cocycle for trivial group action

A 2-cocycle is a function $f:G\times G\to A$ such that:

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

In particular, a 2-cocycle for the trivial group action is a function $f:G\times G\to A$ such that:

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