Manuscripta Mathematica | 2021
Homotopy nilpotency of some homogeneous spaces
Abstract
Let $${\\mathbb {K}}={\\mathbb {R}},\\,{\\mathbb {C}}$$\n , the field of reals or complex numbers and $${\\mathbb {H}}$$\n , the skew $${\\mathbb {R}}$$\n -algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\\Omega (G_{n,m}({\\mathbb {K}}))$$\n , $$\\Omega (F_{n;n_1,\\ldots ,n_k}({\\mathbb {K}}))$$\n , and $$\\Omega (V_{n,m}({\\mathbb {K}}))$$\n of Grassmann $$G_{n,m}({\\mathbb {K}})$$\n , flag $$F_{n;n_1,\\ldots ,n_k}({\\mathbb {K}})$$\n and Stiefel $$V_{n,m}({\\mathbb {K}})$$\n manifolds. Additionally, homotopy nilpotency classes of p-localized $$\\Omega (G^+_{n,m}({\\mathbb {K}})_{(p)})$$\n and $$\\Omega (V_{n,m}({\\mathbb {K}})_{(p)})$$\n for certain primes p are estimated, where $$G^+_{n,m}({\\mathbb {K}})_{(p)}$$\n 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.