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# Homology, Homotopy and Applications

## Volume 25 (2023)

### Number 2

### The homotopy solvability of compact Lie groups and homogenous topological spaces

Pages: 75 – 95

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n2.a5

#### Author

#### Abstract

$\def\F{\mathbb{F}} \def\O{\mathbb{O}} \def\R{\mathbb{R}} \def\C{\mathbb{C}} \def\H{\mathbb{H}}$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.

Received 27 April 2022

Received revised 14 September 2022

Accepted 14 September 2022

Published 4 October 2023