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Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle $\omega_3^G$ of a cohomology group $H^3[G,\mathbb{R}/\mathbb{Z}]$ for the case when $G$ is a continuous group, say $G=U(1)=SO(2)$, instead of a finite group.

We already know that:

$$\omega_3^G \in H^3[G,\mathbb{R}/\mathbb{Z}]=\mathbb{Z}$$

Consider a $3$-cocycle $\omega_3^G \in H^3(U(1),\mathbb{R}/\mathbb{Z})=\mathbb{Z}$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. There are $\mathbb{Z}$ classes of $\omega_3^G$.

Question: We ask whether there exists some group $N$ as a normal subgroup of some bigger group $J$, such that $G$ is the quotient group $$ \frac{J}{N}=G $$ and such that we can always trivialize the $3$-cocycle $\omega_3^G$ of $G=U(1)$ into $3$-coboundary if we lift $G$ into a larger group $J$? Given that we know the group homomorphism $r$: $$J \overset{r}{\rightarrow} G.$$ Namely, $$ \omega_3^G(g_i,\dots)=\omega_3^G(r(j_i),\dots)= \delta \beta_{2}^J(j_i,\dots). $$ with $g \in G, j \in J$. Here $\beta_{2}^J$ is a 2-cochain of $J$.

Guess: I am guessing that we may not be able to find such a $J$ to trivialize the 3-cocycle of $G=U(1)$. How can we verify or prove this?

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