# 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 $$G=SO(3)$$