Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula
$$
\sigma(x)\sigma(y)\alpha(x,y)=\sigma(xy),
$$
for all $x,y\in G/N$.


Then $\alpha$ records how far $\sigma$ is from being multiplicative. Is there a standard name for $\alpha$? It seems to be a variant of a $2$-cocycle, but it is not the usual $2$-cocylce associated to a cross section.



For a cross section $\sigma\colon G/N\to G$, how is $\sigma(y)^{-1}\sigma(x)^{-1}\sigma(xy)$ called?
I believe David Roberts is correct that $\alpha$ is a $2$-cocycle representing an element of $H^2(G/N,N)$, and is the usual $2$-cocycle associated to this extension.


The confusion comes from the fact that $\omega$ is *not* a $2$-cocycle. Rather, it is called a cocycle because it can be thought of as a $1$-cocycle representing a class in $H^1 \bigl( G, \mathcal{F}(G/N; N) \bigr)$, where $\mathcal{F}(G/N; N)$ is the space of functions from $G/N$ to $N$. This is a group under pointwise multiplication, and $G$ acts on it via $\varphi^g(x) = \varphi(gx)$.


More precisely, define $\omega\_g(x) = \omega(g,x)$, so $\omega\_g \in \mathcal{F}(G/N; N)$ for each $g \in G$. Then 
 $$ \sigma(ghx) \, \omega(gh,x) 
 = gh \, \sigma(x) 
 = g \cdot \sigma(hx) \, \omega(h,x)
 = \sigma(ghx) \, \omega(g, hx) \, \omega(h,x)
 ,$$
so $\omega\_{gh} = (\omega\_g)^h \cdot \omega\_h$. This is precisely what it means to say that $\omega\_g$ is a $1$-cocycle. 


That $\omega$ is a $1$-cocycle (in the more general context of $G$-actions on principal bundles) is mentioned at the bottom of page 66 of R.J.Zimmer's book *Ergodic Theory and Semisimple Groups*.


