# High-dimensional analogue of Skew

Suppose G and A are abelian groups. Suppose f is a 2-cocycle for the trivial group action of G on A. In other words, we have that:

$$f(g_1,g_2 + g_3) + f(g_2,g_3) = f(g_1 + g_2,g_3) + f(g_1,g_2)$$

for all $g_1,g_2,g_3 \in G$. Then we can show that the map:

$$\operatorname{Skew}(f) = (g_1,g_2) \mapsto f(g_1,g_2) - f(g_2,g_1)$$

is an alternating biadditive map from $G \times G$ to $A$. The proof is straightforward but doesn't seem entirely obvious. This is related to the fact that, in a group of nilpotency class two, the commutator of two elements is a homomorphism in each input.

It is true that any function $G^n \to A$ that is additive in each of the inputs is a n-cocycle. Question: Is there some analogue of Skew for higher n that starts with an arbitrary n-cocycle and outputs a map that's additive in each coordinate? Ideally, the map should have the additional property that when applied a second time, it acts just like multiplication by n.

In the Skew case, all the outputs are additionally restricted to being alternating; a similar restriction in the general case is fine.

If such maps do not exist for higher n, is there an easy reason/explanation for the fact?

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