Take a compact connected simple centreless Lie group $G$. Can the

commutator map$G\times G\to G$ sending $(x,y)$ to $[x,y]$ be homotopic to a constant map?

I am interested mostly in the case, where $G={\rm PSU}(n)$.

As far as I understand, the commutator map is homologically trivial (right?).

There is a related question.