TITLE: What is π_1(BG) for an arbitrary topological group $G$?
QUESTION [16 upvotes]: 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.

REPLY [2 votes]: Readers might also like to know a result on the first k-invariant of BG contained in
R. Brown and C.B. Spencer,  $\cal G$-groupoids, crossed modules
and the  fundamental groupoid of a topological group'', Proc.
Kon. Ned. Akad. v.  Wet. 7 (1976) 296-302.
This shows that the associated crossed module to fundamental groupoid of $G$ has   $k$-invariant which is exactly the $k$-invariant of $BG$. 
Since this query is about non-connected topological groups, another related ressult is on universal covers of non-connected topological groups 
R. Brown and O. Mucuk, ``Covering groups of non-connected
topological  groups revisited'',  Math. Proc. Camb. Phil.
Soc,  115 (1994) 97-110.
which relates the question of the existence of topological group universal covers of $G$ to the theory of ostructions to extensions of abstract groups.