Homotopy nilpotency of some homogeneous spaces

Abstract

Let {mathbb {K}}={mathbb {R}},,{mathbb {C}}, the field of reals or complex numbers and {mathbb {H}}, the skew {mathbb {R}}-algebra of quaternions. We study the homotopy nilpotency of the loop spaces Omega (G_{n,m}({mathbb {K}})), Omega (F_{n;n_1,ldots ,n_k}({mathbb {K}})), and Omega (V_{n,m}({mathbb {K}})) of Grassmann G_{n,m}({mathbb {K}}), flag F_{n;n_1,ldots ,n_k}({mathbb {K}}) and Stiefel V_{n,m}({mathbb {K}}) manifolds. Additionally, homotopy nilpotency classes of p-localized Omega (G^+_{n,m}({mathbb {K}})_{(p)}) and Omega (V_{n,m}({mathbb {K}})_{(p)}) for certain primes p are estimated, where G^+_{n,m}({mathbb {K}})_{(p)} is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.