Hi,


It is well known that the fundamental group of a topological group is abelian, and that every group is the fundamental group of some topological space.


My question is: Does every abelian group arise as the fundamental group of some topological group or are there other restrictions?



Abelian groups as fundamental groups of topological groups
Yes, if $G$ is an abelian group, its classifying space $BG$ is an *abelian* topological group whose $\pi \_1$ is $G$. You can find details in John Baez's wonderful post, hearteningly called "Classifying Spaces Made Easy"


<http://math.ucr.edu/home/baez/calgary/BG.html> 


or in the answer by Chris Schommer-Priess to the following question on this site:


[Classifying Space of a Group Extension](https://mathoverflow.net/questions/12144/classifying-space-of-a-group-extension) 


