Parallelizability of homogeneous spaces. II
type: Item
instance of: scholarly article
title: Parallelizability of homogeneous spaces. II
zbMATH Open document ID: 0572.57014
DOI: 10.1007/BF01458022
author: Wilhelm Singhof
author: Dieter Wemmer
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164092
review text: [For part I see ibid. 260, 101-116 (1982; Zbl 0471.57005).] A complete classification of the parallelizable and stably parallelizable homogeneous spaces G/H is obtained when G is a simple 1-connected compact Lie group and H a closed connected subgroup of G. As a consequence one gets the following results: (1) G/H is stably parallelizable if and only if the adjoint representation \(Ad_ H\) of H is contained in the image of the restriction map of real representation rings RO(G)\(\to RO(H)\). (2) Assume that G/H is stably parallelizable and not parallelizable. Then G/H is a sphere of rank G\(=rank H.\) The main tools are non-vanishing criteria for characteristic classes based on the collapsing of the Eilenberg-Moore spectral sequence and the injectivity of the natural homomorphism \(\alpha\) : R(H)\(\oplus_{R(G)}{\mathbb{Z}}\to K(G/H)\) established by V. P. Snaith. By these methods, the problem is reduced to the study of certain concrete situations which are treated individually.
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57T35
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55T20
zbMATH DE Number: 3913286
zbMATH Keywords: parallelizable homogeneous spaces
zbMATH Keywords: compact Lie group
zbMATH Keywords: adjoint representation
zbMATH Keywords: real representation rings
zbMATH Keywords: stably parallelizable
zbMATH Keywords: characteristic classes
zbMATH Keywords: Eilenberg-Moore spectral sequence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3913286