TITLE: Are compact simple groups homotopically non-abelian?
QUESTION [9 upvotes]: 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.

REPLY [7 votes]: 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.