A survey of homotopy nilpotency and co-nilpotency

Abstract

We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces.Next, we take up the systematic study of homotopy nilpotency of homogenous spaces $G/K$ for a Lie group $G$ and its closed subgroup $K<G$.The homotopy nilpotency of the loop spaces $\Omega (G_{n,m}(\mathbb{K}))$ and $\Omega (V_{n,m}(\mathbb{K}))$ of Grassmann $G_{n,m}(\mathbb{K})$and Stiefel $V_{n,m}(\mathbb{K})$ manifolds for $\mathbb{K}=\mathbb{R},\,\mathbb{C}$, the field of reals or complex numbers and $\mathbb{H}$, the skew$\mathbb{R}$-algebra of quaternions is shown.

On homotopy nilpotency of some suspended spaces

“…Let S 2m´1 (p) be the p-localization of the sphere S 2m´1 at a prime p. The main result of the paper [12] is the explicit determination of the homotopy nilpotence class of a wide range of homotopy associative multiplications on localized spheres S 2m´1 (p) for p ą 3. Furthermore, the paper [11] reviews known and states some new results on the homotopy nilpotency and conilpotency of spaces.…”