It is well-known that if $G$ is a discrete group, then $BG=K(G,1)$. I'm interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups but equipped with the discrete topology. They will be quite different in general, that much is clear.

For instance, $S^1$ can be thought of as a topological group (the usual way) and as a discrete group, denoted by, say, $S^1_d$. On one hand we have that $BS^1=\mathbb{C}P^∞(=K(\mathbb{Z},2))$, and on the other we should have $K(S^1_d,1)$, which I guess we could construct by hand (as in Hatcher's book, for example), but other than that I can't really say anything about it. Perhaps they usually are quite messy.

What is known about this space? Or, more generally, about $K(G,1)$ where $G$ is infinite and discrete (and not $\mathbb{Z}$)? Are there any references about this?

Any thoughts on the relation between the classifying spaces of top. groups vs E-M spaces of the same (discrete) groups (if any) would be greatly appreciated.

Thanks!

On another matter: this is a cross-post of https://math.stackexchange.com/questions/480437/what-is-k-s1-1, which I asked quite some time ago, and didn't get any answers there. Should I delete that question?

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