Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a slightly more complicated but naturally-occuring map: $(g,h)\mapsto ghg^{-1}h^{-1}$, i.e. $G\times G\to [G,G]\hookrightarrow G$. What goes on at the fundamental group level? 


In other words, **is it true that $\pi\_1(G\times G)\to\pi\_1([G,G])\to\pi\_1(G)$ is exact?** 


I have a rather ad hoc reason to believe that the first map is trivial (as $\pi\_1$ is abelian here, the commutator $[g,h]$ will unwind itself to the constant loop) and so I would want the second map to be injective.


*Update*: The comments below take care of this when $G$ is a Lie group!  

So what can obstruct $\pi\_1$ being injective on $[G,G]\hookrightarrow G$ for non-Lie groups?  

*Update*: It has also been pointed out that this works for finite-dimensional topological groups!  

That leaves a possible counterexample for the infinite-dimensional case.



$\pi_1$ Sequence of Topological Groups
This is really more of a comment, but it kind of answers one of the OP's question, so I am indulging myself: It is a result of W. Browder (Annals, 1961) that $\pi\_2$ of a *finite dimensional* $H$-space is trivial, so the result holds in that setting. I learned of this (and also that this is not true without the finite dimensionality assumption) from @Allen Hatcher's answer to this (very relevant) question:
[Homotopy groups of Lie groups](https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups)


