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?



Eilenberg-MacLane Spaces of "large" groups
Let me just point out that if you're interested in, say, homology, then discrete $S^{1}$ is not as complicated as it might seem. The resulting invariants will be huge, of course, but one should be able to compute them explicitly. 


The point is that $S^{1} = \mathbb{R} / \mathbb{Z}$ and $\mathbb{R} \simeq \bigoplus \mathbb{Q}$ (as an abelian group), thus $S^{1} \simeq (\mathbb{Q} / \mathbb{Z}) \bigoplus (\oplus \mathbb{Q})$. Any direct sum of groups can be written as a filtered colimit of its finite subsums and finite direct sums coincide with finite products. 


Taking classifying spaces commutes with both filtered colimits and finite products and so the classifying space of discrete $S^{1}$ can be described in terms of classifying spaces of $\mathbb{Q} / \mathbb{Z}$ and $\mathbb{Q}$. You can use this to compute homology (it commutes with filtered colimits) by using Künneth formula to deal with products.


I imagine classifying spaces of $\mathbb{Q} / \mathbb{Z}$ and $\mathbb{Q}$ are not difficult to describe, as $\mathbb{Q} / \mathbb{Z}$ falls apart into a direct sum of $p$-torsion parts (which I imagine are limits of finite cyclic $p$-groups?) and $\mathbb{Q} = colim (\mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \ldots)$. 


