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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?

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2 Answers

up vote 16 down vote accepted

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

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I'd like to add a quick little explanation to Georges's already sufficient answer (I'm sure this explanation is in Baez's post, but it can be said in a few lines here).

One of the morals of the famous Eckmann-Hilton lemma is that an abelian group is an abelian group object in the category of groups. Now, the classifying space functor $B: Grp \to Top$ preserves products (here $Top$ should be a convenient category of spaces like compactly generated weak Hausdorff spaces; see nLab). But product-preserving functors take algebraic gizmos (like abelian group objects) to algebraic gizmos. So $B$ takes abelian groups $G$ = abelian group objects in $Grp$ to abelian group objects in $Top$, i.e., topological abelian groups. And since we have a universal covering fibration $G \to EG \to BG$, we get $\pi_1(BG) \cong G$.

A refinement of this observation shows that we have a classifying space functor

$$B: TopAb \to TopAb$$

which can be used to realize topological abelian groups as infinite loop spaces.

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