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Groupprops β

Cocycle for a group action



Suppose G is a group and A is an abelian group, with an action \varphi of G on A. In other words, \varphi is a homomorphism of groups from G to \operatorname{Aut}(A), the automorphism group of 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:

\! \varphi(g_1)(f(g_2,g_3,\dots,g_{n+1})) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n + 1}f(g_1,g_2,\dots,g_n) = 0

If we suppress the symbol \varphi and denote the action by \cdot, this becomes:

\! 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_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} 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_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0

Particular cases

n Condition for being a n-cocycle Further information
1 For all g_1,g_2\in G, we have \! g_1 \cdot f(g_2) - f(g_1g_2) + f(g_1) = 0, equivalently \! f(g_1g_2) = f(g_1) + g_1 \cdot f(g_2) 1-cocycle for a group action
2 For all g_1,g_2,g_3\in G, we have \! g_1 \cdot f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0, equivalently g_1 \cdot f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_2,g_3) 2-cocycle for a group action
3 For all g_1,g_2,g_3,g_4 \in G, we have \!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) 3-cocycle for a group action