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Transitive actions on highly connected spaces


Author: Victor Schneider
Journal: Proc. Amer. Math. Soc. 38 (1973), 179-185
MSC: Primary 57E15
DOI: https://doi.org/10.1090/S0002-9939-1973-0321125-9
MathSciNet review: 0321125
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Abstract: Let $ G$ be a compact, connected Lie group and $ H$ a closed subgroup of $ G$. It is shown that if $ G/H$ is highly connected relative to $ \operatorname{Rk} (G) - \operatorname{Rk} (H),G/H$ splits as a product of homogeneous spaces of simple Lie groups. This is used to show that the only transitive, effective actions on a large class of products of spheres are products of the known actions on the individual spheres.


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  • [1] A. Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris 230 (1950), 1378-1380. MR 11, 640. MR 0034768 (11:640c)
  • [2] -, Sur l'homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math. 76 (1954), 273-342. MR 16, 219. MR 0064056 (16:219b)
  • [3] -, Topology of Lie groups and characteristic classes, Bull. Amer. Math. Soc. 61 (1955), 397-432. MR 17, 282. MR 0072426 (17:282b)
  • [4] A. Borel and J. de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200-221. MR 11, 326. MR 0032659 (11:326d)
  • [5] W. Y. Hsiang and J. C. Su, On the classification of transitive effective actions on Stiefel manifolds, Trans. Amer. Math. Soc. 130 (1968), 322-336. MR 36 #4581. MR 0221529 (36:4581)
  • [6] M. Mimura, On the homotopy groups of the exceptional Lie groups, Conference on Algebraic Topology, University of Illinois, Urbana, Ill., 1968.
  • [7] D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. (2) 44 (1943), 454-470. MR 5, 60. MR 0008817 (5:60b)
  • [8] A. Oniščik, Transitive compact transformation groups, Mat. Sb. 60 (102) (1963), 447-485; English transl., Amer. Math. Soc. Transl. (2) 55 (1966), 153-194. MR 27 #5868. MR 0155935 (27:5868)
  • [9] J. Poncet, Groupes de Lie compacts de transformations de l'espace euclidian et les sphères comme espaces homogènes, Comment. Math. Helv. 33 (1959), 109-120. MR 21 #2708. MR 0103946 (21:2708)
  • [10] H. Scheerer, Homotopieäquivalente kompakte Liesche Gruppen, Topology 7 (1968), 227-232. MR 37 #4833. MR 0229259 (37:4833)
  • [11] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, no. 49, Princeton Univ. Press, Princeton, N.J., 1962. MR 26 #777. MR 0143217 (26:777)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0321125-9
Keywords: Transitive action, $ n$-connected, Lie group, Steenrod operations
Article copyright: © Copyright 1973 American Mathematical Society

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