Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This question already has an answer here:

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I think, it is still true if we do not suppose path-connectedness. Right?

I do not know homotopy. I have just learned the fundamental group.

share|cite|improve this question

marked as duplicate by Daniel Fischer Mar 5 at 21:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

The fundamental group depends only on the path component of the base point, so the path-connectedness just says that there are no irrelevant path components, it's nothing essential. –  Daniel Fischer Feb 22 '14 at 23:03
Thank you very much! –  square Feb 22 '14 at 23:11
How can one know what the fundamental group is without knowing some basic results on homotopy? –  PVAL Feb 23 '14 at 0:10
PVAL I know some basic results on homotopy. I have studied chapter0 and beginning of chapter1 Hatcher. I mean I have not studied chapter4. –  square Mar 4 '14 at 1:53

1 Answer 1

The usual proof is to show that for all loops $\alpha,\beta \colon [0,1] \to G$ of the topological group $G$, the concatenation $\alpha \cdot \beta$ is homotop to $t \mapsto \alpha(t)\beta(t)$ and to $t \mapsto \beta(t)\alpha(t)$. It requires to exhibit formulae.

Howerver, my favourite proof of this result is the following one, from Grothendieck (I think) : the fundamental group functor $\pi_1 \colon \mathsf{pcTop} \to \mathsf{Grp}$ from the category of path-connected topological spaces to the category of groups respects products (classical lemma), so sends group objects to group objects ; the group objects of $\mathsf{pcTop}$, which are the path-connected topological groups (by definition), are send to group objects of $\mathsf{Grp}$, which are the abelian groups (easy exercise).

share|cite|improve this answer
Hey, cute abstract nonsense proof! Thanks for posting it! (+1) –  lentic catachresis Feb 23 '14 at 9:41
Thanks a lot. @Pece –  square Mar 4 '14 at 1:56
@lenticcatachresis I like the cute in "cute abstract nonsense proof". You're also absolutely right. –  Joachim Jan 10 at 19:16
Actually, you should define the fundamental group functor on the category of based spaces. Otherwise you can't even define the action on morphisms. –  Zhen Lin Mar 5 at 16:02

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