The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\pi\_{1}(BG)$? A reference on this would be great. My initial guess: $\pi\_{1}(BG)$ is the quotient group $\pi\_{0}(G)$ for arbitrary $G$


Motivation: There is a natural way to make $\pi\_1$ a functor to topological groups. I am interested in relating the topologies of $G$ and $\pi\_{1}(BG)$ but the topology on $\pi\_{1}(X)$ is boring (discrete) when $X$ is a CW-complex.



What is π_1(BG) for an arbitrary topological group $G$?
If $G$ is homeomorphic to a Cantor set (e.g. $G=\mathbb Z\_p$), then $BG$ contains a copy of the Hawaiian earrings in it. To see this, take a sequence of points of $G$ that converges to the identity element: you'll get a corresponding sequence of 1-cells in $BG$ that converge to the the degenerate 1-cell. The fundamental group of the Hawaiian earrings is a rather wild object, and looks nothing like the free group that you might naively expect.  



If, on the other hand, if you agreed to redefine $BG$ to be the **fat** geometric realization of the simplicial space $NG$, then you would get $\pi\_1(BG)\cong\pi\_0(G)$, as desired.
I would even bet that the above isomorphism respects the natural topologies that are present on both sides.


