
> 
> 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 $G={\rm PSU}(n)$.




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


There is a [related question](https://mathoverflow.net/questions/45483/word-maps-on-compact-lie-groups).



Are compact simple groups homotopically non-abelian?
The following is the main theorem in 


Araki, S.; James, I. M.; Thomas, E., *"Homotopy-abelian Lie groups"*, Bull. Amer. Math. Soc. 1960.



> 
> **Theorem:** A compact connected Lie group is homotopy-abelian only if it is abelian.
> 
> 
> 


