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 Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is shown.

Keywords

• classical lie group
• h -fibration
• flag manifold
• (oriented) grassmann manifold
• homogenous space
• homology decomposition
• homotopy nilpotency (co-nilpotency) class
• h -space
• h0-space
• n-fold commutator (co-commutator) map
• p-localization
• loop space
• morava k-theory
• postnikov system
• projective space
• sphere
• stiefel manifold
• suspension space
• smash product