proof by nuke of the fact that fundamental group of topological group is abelian

Question

"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton argument.But can some kind of sledgehammer be used to give a direct proof.

Answer 1

Consider the classifying space $BG$ of $G$. One has the fibration $G \hookrightarrow EG \to BG,$ and passing to the associated long exact sequence of homotopy groups, and using the contractibility of $EG$, we find that $\pi_1(G) \cong \pi_2(BG).$ Since $\pi_2$ is abelian, we see that $\pi_1(G)$ is abelian.