gr.group theory – Typical preimage of the commutator map


By Goto’s theorem for any compact connected semisimple Lie group $G$ of dimension $n$, any element $xin G$ is a commutator, namely $x=(y,z)$ for some $y, zin G$. Another way to say it is that the commutator map $pi:Gtimes Grightarrow G$ is surjective. By Sard’s Lemma it follows that typical element $win G$ is a regular value of $pi$ and ${pi}^{-1}(w)subset Gtimes G$ is a smooth compact submanifold of dimension $n$.

Question: what is the homeomorphic type of this manifold for typical $w$?

Of course it is tempting to suspect that ${pi}^{-1}(w)$ is homeomorphic to $G$ but somehow I have difficulty in checking it even in rather simple case of $G=SO(3)$