Homotopy nilpotency of some homogeneous spaces

Let $${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$ K = R , C , the field of reals or complex numbers and $${\mathbb {H}}$$ H , the skew $${\mathbb {R}}$$ R -algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$ Ω ( G n , m ( K ) ) , $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$ Ω ( F n ; n 1 , … , n k ( K ) ) , and $$\Omega (V_{n,m}({\mathbb {K}}))$$ Ω ( V n , m ( K ) ) of Grassmann $$G_{n,m}({\mathbb {K}})$$ G n , m ( K ) , flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$ F n ; n 1 , … , n k ( K ) and Stiefel $$V_{n,m}({\mathbb {K}})$$ V n , m ( K ) manifolds. Additionally, homotopy nilpotency classes of p-localized $$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( G n , m + ( K ) ( p ) ) and $$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( V n , m ( K ) ( p ) ) for certain primes p are estimated, where $$G^+_{n,m}({\mathbb {K}})_{(p)}$$ G n , m + ( 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.

Topological complexity of real Grassmannians

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds G k (ℝ n ) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝ n . In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of G k (ℝ n ) as a function of n.