Abstract
AbstractGanea proved that the loop space of
$\mathbb{C} P^n$
is homotopy commutative if and only if
$n=3$
. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but
$\mathbb{C} P^3$
are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds
$G/T$
for a maximal torus T of a compact, connected Lie group G.
Publisher
Cambridge University Press (CUP)