The following theorem is proved in Milnor's famous book "Morse theory".
Theorem 21.7 (Bott). Let $G$ be a compact, simply connected Lie group. Then the loop space of $G$ has the homotopy type of a CW-complex with no odd dimensional cells.
It is not clear to me where the author uses the simply connectedness of $G$. Is it a necessary condition? Can someone please illuminate?