Parallelizability of homogeneous spaces. II (Q1062320)

[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.