@Urs: That's not the continuous cohomology from the reference. The continuous cohomology is the cohomology of the cochain complex of continuous functions and the standard group differential. This vanishes for a (simply connected) compact group (with connected coefficients). But the cohomology you are referring to is for compact groups and U(1)-coefficients the cohomology of the classifying space (with a degree shift).

Sorry, there are too many $H^n$ flying around. Let us call $H^n_{t}(G,A)$ the topological group cohomology and $H^n(BG,A)$ the cohomology of the classifying space with discrete coefficients $A$. What we have are the isomorphisms $H^n(BG,Z_m)\cong H^n_B(G,Z_m)\cong H^{n+1}_B(G,A)$ for $A$ as described (and $G$ locally compact and finite-dimensional), and also $H^n(BG,Z_m)\cong H^n_t(G,Z_m)\cong H^{n+1}_t(G,A)$. Could you now repeat your question?

@Xiao-Gang: Yes, this runs under the name "dimension shifting". Take $E Z_n$ the group of (equivalence classes of) measurable $Z_n$-valued functions on the unit interval. With the distance in measure this should be a contractible polish group. Then take $C(G,E Z_n)$, which is still contractible and polish. Moreover, it is soft, so the topological group cohomology (and under the above assumption on $G$) of it vanishes (see the appendix in our paper). Since $Z_n$ is finite, it acts properly discontinuously on $C(G,E Z_n)$ and so $E:=C(G,E Z_n)\to C(G,E Z_n)/Z_n=:A$ has a continuous section.

Our results work also for finite-dimensional locally compact groups. The problems with $Z_n$-coefficients would be to find a short exact sequence $Z_n \to E \to A$ where you see that $H^n_B(G,E)$ vanishes. (To this sequence you would like to apply the long exact sequence to in oder to identify $H^n_B(G,Z_n)$ with $H^n_B(G,A)$ in case $H^n_B(G,E)$ vanishes. For $Z$ you can take $Z \to R \to U(1)$.)

Hi Uli! A finite-dimensional Lie group is rationally a product of odd spheres, so I wouldn't expect a reasonable group structure on $G_\mathbb{Q}$ (although this is far from being a no-go argument)...