3-cocycle for a group action

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This article gives a basic definition in the following area: group cohomology
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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


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_is 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.