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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 Joan 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