# Classifying spaces of a profinite groups

As a topological group, a profinite group $G$ has a classifying space $BG$. On the other hand, since $G = \underleftarrow{\lim}\; G/U\;\;(U \le G$ open$)$ is an inverse limit of finite groups, we also have an inverse system $B(G/U)$ of topological spaces and can form the space $\underleftarrow{\lim}B(G/U)$.

Question 1: How do $BG$ and $\underleftarrow{\lim}B(G/U)$ relate ? Can we, for example say, that $\underleftarrow{\lim}B(G/U)$ is a classifying space for $G$ ?

Question 2: Are there futher results about classifying spaces of profinite groups ?

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Even if it does not directly pertain to your question, I think the following would be good to look at: Gereon Quick sets up some good technology for working with profinite groups. He also makes it seem easy. math.harvard.edu/~gquick/HomfixedpointsforLT03182012.pdf His other work might be of interest to you as well. – Sean Tilson May 3 '12 at 21:08

There's a lot of literature on classifying spaces of the $p$-adic integers. For instance, there are two 45-year old "computations" of the complex K-theory of the $p$-adic integers, resulting in two different answers. An explanation of why they are different, only found recently, is by considering a more refined notion of a classifying space, which is a uniform space well-defined up to uniform homotopy equivalence (see, for lack of a better account, subsection 1.4 here).