Abstract
In this paper we investigate transitive actions of compact connected Lie groups on certain spaces X which are not spheres, whose dimension is not too small and whose rational cohomology algebra is an exterior algebra on homogeneous generators of odd degree. In case X is a simply connected classical group, a 3-connected real or a 5-connected complex or a quaternionic Stiefel manifold, we obtain (in principle) the classification of the transitive actions on X up to equivariant homeomorphism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
BOREL, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. 57, 115–207 (1953).
BOREL, A.: Le plan projectif des octaves et les sphéres comme espaces homogènes. Compt. Rend. 230, 1378–1380 (1950).
BOREL, A.; La cohomologie mod 2 de certaines espaces homogènes. Comm. Math. Helv. 27, 165–197 (1953).
BOREL, A.: Topology of Lie groups and characteristic classes. Bull. Am. Math. Soc. 61, 397–432 (1955).
BROWDER, W.: Fiberings of spheres and H-spaces which are rational homology spheres. Bull. Am. Math. Soc. 68, 202–203 (1962).
DOLD, A.: Űber fasernweise Homotopieäquivalenz von Faserräumen. Math. Z. 62, 111–126 (1955).
DYNKIN, E.B.: Semisimple subalgebras of semisimple algebras. Transl. Am. Math. Soc. (2) 6, 111–244 (1957).
DYNKIN, E.B.: Maximal subgroups of the classical groups. Transl. Am. Math. Soc. (2) 6, 245–378 (1957).
DYNKIN, E.B.: Topological characteristics of homomorphism of Lie groups. Transl. Am. Math. Soc. (2) 12, 301–342 (1959).
HARRIS, B.: Torsion in Lie groups and related spaces. Top. 5, 347–354 (1966).
HSIANG, W-Y.: A survey on regularity theorems in differentiable compact transformation groups. Proc. Conf. transf. groups Proceedings of the Conference on Transformation Groups New Orleans 1967. Berlin-Heidelberg-New York: Springer 1968, 77–124.
HSIANG, W-Y., SU, J. C.: On the classification transitive actions on Stiefold manifolds. Trans. Am. Math. Soc. 130, 322–336 (1968).
JAMES, I.M.: Note on the homotopy type of Stiefel manifolds. To appear.
JAMES, I.M.: On the decomposabiltiy of fibre spaces. Proc. Steenrod Birthday Conf. To appear.
ONISCIK, A.L.: Inclusion relations among transitive compact transformation groups. Transl. Am. Math. Soc. (2) 50, 5–58 (1966).
ONISCIK, A.L.: Transitive compact transformation groups. Transl. Am. Math. Soc. (2) 55, 153–195 (1966).
SUTER, U.: Die Nicht-Existenz von Schnittflächen komplexer Stiefel-Mannigfaltigkeiten. Math. Z. 113, 196–204 (1970).
TROST, E.: Primzahlen. Basel-Stuttgart: Birkhäuser 1953.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Scheerer, H. Transitive actions on Hopf homogeneous spaces. Manuscripta Math 4, 99–134 (1971). https://doi.org/10.1007/BF01169407
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01169407