The homotopy solvability of compact Lie groups and homogenous topological spaces

$\def\F{\mathbb{F}} \def\O{\mathbb{O}} \def\R{\mathbb{R}} \def\C{\mathbb{C}} \def\H{\mathbb{H}}$We analyse the homotopy solvability of the classical Lie groups $O(n)$, $U(n)$, $Sp(n)$ and derive its heredity by closed subgroups. In particular, the homotopy solvability of compact Lie groups is shown.

Then, we study the homotopy solvability of the loop spaces $\Omega (G_{n,m} (\F))$, $\Omega (V_{n,m} (\F))$ and $\Omega (F_{n; n_1,\dotsc,n_k}(\F))$ for Grassmann $G_{n,m} (\F)$, Stiefel $V_{n,m} (\F)$ and generalised flag $F_{n; n_1,\dotsc,n_k}(\F)$ manifolds for $\F = \R, \C$, the field of reals or complex numbers and $\H$, the skew $\R$-algebra of quaternions. Furthermore, the homotopy solvability of the loop space $\Omega (\O P^2)$ for the Cayley plane $\O P^2$ is established as well.

Keywords

Cayley plane, Grassmann (generalised flag and Stiefel) manifold, $H$-space, localization, $n$-fold commutator map, nilpotent space, nilpotency (solvability) class, loop space, Postnikov system, Samelson product, smash product, suspension space, wedge sum, Whitehead product

2010 Mathematics Subject Classification

Primary 55P15. Secondary 14M17, 22C05, 55P45, 55R35.

Full Text (PDF format)

Received 27 April 2022

Received revised 14 September 2022

Accepted 14 September 2022

Published 4 October 2023