In Juven Wang, Zheng-Cheng Gu, and Xiao-Gang Wen – Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, the authors calculated many group cocycles in explicit form. See n-cocycles of finite abelian groups from cohomology group for more discussions. In all of those examples, the group cocycle is always additive with respect to at least 1 argument, e.g.
$$\omega(a,b,c)=\exp\Big(\frac{2\pi ip}{N^{2}}a(b+c-[b+c]_{N})\Big)$$
(for some 3-cocyles in $H^{3}(\mathbb{Z}_{N};U(1))$) where
$$[b+c]_{N}=\begin{cases}b+c, & 0\leq b+c<N\\
b+c-N, & b+c\geq N.
\end{cases}$$
Similarly, for some 4-cocycles in $H^{4}(\mathbb{Z}_{N_{1}}\times\mathbb{Z}_{N_{2}};U(1))$
$$\omega((a_{1},a_{2}),\dotsc,(d_{1},d_{2}))=\exp\Big(\frac{2\pi ip_{II(12)}}{N_{12}N_{2}}a_{1}b_{2}(c_{2}+d_{2}-[c_{2}+d_{2}]_{N_{2}}\Big)$$
where $N_{12}:=\gcd(N_{1},N_{2})$, see the first ref above for more examples.
The additivity is not so obvious in the first place (since group cochains are not homomorphism in general).
Is there a simple reason for cocycles to be additive (w.r.t. say, the first argument)? I didn't find an explanation in my homological algebra textbook (and I am sorry if I missed something obvious because I am a physicist).
(I am also wondering how general the additivity is.)
Best Answer
There are plenty of cocycles which are not additive in any variable, and plenty of cohomology classes that do not admit additive representatives. For example, if $G$ is finite, then there are no [nontrivial] additive functions $G \to \mathbb{Z}$, and so no [nontrivial] $\mathbb{Z}$-valued cohomology class has an additive (in any variable) representative.
What's happening in your examples is that the class in question is a cup product. In the first case, you have the group $G = \mathbb{Z}_N$, and recall that as a ring $\operatorname{H}^\bullet(G; \mathbb{Z}) = \mathbb{Z}[\phi]/N\phi$, where $\phi \in \operatorname{H}^2$ is the generator. Recall also that the Bockstein map $\beta : \operatorname{H}^{\bullet-1}(G; U(1)) \to \operatorname{H}^\bullet(G; \mathbb{Z})$ is an isomorphism, and that $\beta^{-1}\phi \in \operatorname{H}^{1}(G; U(1)) = \hom(G, U(1))$ is the standard inclusion $a \mapsto \exp(2\pi i a / N)$. But the Bockstein map is $\operatorname{H}^\bullet(G; \mathbb{Z})$-linear, and so $\omega = \beta^{-1}(\phi^2) = \phi \cup \beta^{-1}(\phi)$. The additivity that you observe follows from the additivity of $\beta^{-1}(\phi)$.
The second case is similar: the class you care about is a cup product of a class in $\operatorname{H}^2(G; \mathbb{Z})$ with a class in $\operatorname{H}^2(G; U(1))$. This latter class comes from a bihomomorphism.
In general, multihomomorphisms do supply cocycles. Usually the map from multihomomorphisms to cohomology classes is neither injective nor surjective, but it sometimes is, especially in low degree and with specific coefficients. For example, $\operatorname{H}^1(G;A) = \hom(G,A)$ is always a space of homomorphisms. In degree 2, the Künneth formula shows that for $G$ finite abelian, $\operatorname{H}^2(G; U(1))$ is surjected (but not injected) by the set of bihomomorphisms. If $G$ is an elementary abelian $2$-group, then $\operatorname{H}^\bullet(G; \mathbb{F}_2) = \operatorname{Sym}^\bullet(\hom(G, \mathbb{F}_2))$ manifestly consists entirely of multihomomorphisms. Furthermore, in this case the standard inclusion $\mathbb{F}_2 \hookrightarrow U(1)$ induces a surjection $\operatorname{H}^\bullet(G; \mathbb{F}_2) \to \operatorname{H}^\bullet(G; U(1))$, so again you find that every class is represented by a multihomomorphism. If $G$ is an elementary abelian $p$-group with $p$ odd, then $\operatorname{H}^\bullet(G; \mathbb{F}_p) = \operatorname{Alt}^\bullet(\hom(G, \mathbb{F}_p)) \otimes \operatorname{Sym}^\bullet(\hom(G, \mathbb{F}_p))$, with the $\operatorname{Alt}$ part the subalgebra generated in degree $1$; that $\operatorname{Alt}$ part manifestly consists of multihomomorphisms. This $\operatorname{Alt}$ part injects (via the standard $\mathbb{F}_p \hookrightarrow U(1)$) into the $U(1)$-cohomology, giving you some classes you can easily realize multihomomorphicly. Further cup products then supply cocycles which are partially-multihomomorphic, i.e. additive in some but not all variables.