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.
Best Answer
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