I have recently learnt about group cohomology and I was curious what is the relationship between the different cohomology groups which we can associate to a give topological group $G$. I thought of three different cohomology groups related to $G$:
- group cohomology $H_{grp}^*(G; M)$
- cohomology $H^*(BG; M)$ of the $BG$
- cohomology $H^*(G; M)$ of $G$ as a topological space
How are these related? I have posed my question in terms of groups but I am also interested in the same question in terms of cohomology rings.
By considering $G=S^1,$ so that $BG$ is the infinite-dimensional complex projective space, you can see immediately that there’s no easy relationship between your second two cohomologies. There is actually an excellent relationship here, but it’s in homotopy: the homotopy groups of $BG$ are exactly those of $G,$ shifted up one level. The group cohomology of $G,$ in terms of the standard definition you’ve presumably just learned, is the topological cohomology of the classifying space of the discrete group underlying $G.$ In general, this will be some completely different and uninteresting thing, since it totally ignores the topology on $G$, although there are some particular cases where you want it.