Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


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?

share|improve this question

2 Answers 2

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"


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

Classifying Space of a Group Extension

share|improve this answer

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.