Homogeneous spaces with vanishing Steenrod squaring operations

Schneider, Victor

doi:10.1090/S0002-9939-1975-0405470-6

Homogeneous spaces with vanishing Steenrod squaring operations
HTML articles powered by AMS MathViewer

by Victor Schneider PDF

Proc. Amer. Math. Soc. 50 (1975), 451-458 Request permission

Abstract:

If $G$ is a compact, connected Lie group, $H$ is a closed subgroup of $G$ and $G/H$ has no nonzero Steenrod operations, then $G/H$ splits as a product of homogeneous spaces of simple Lie groups (the factors of $G$). This fact is used to classify transitive actions on spaces with vanishing Steenrod operations, namely product of certain Stiefel manifolds and spheres.

References

Armand Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris 230 (1950), 1378–1380 (French). MR 34768
Armand Borel, Sur l’homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math. 76 (1954), 273–342 (French). MR 64056, DOI 10.2307/2372574
Wu-yi Hsiang and J. C. Su, On the classification of transitive effective actions on Stiefel manifolds, Trans. Amer. Math. Soc. 130 (1968), 322–336. MR 221529, DOI 10.1090/S0002-9947-1968-0221529-1
Deane Montgomery and Hans Samelson, Transformation groups of spheres, Ann. of Math. (2) 44 (1943), 454–470. MR 8817, DOI 10.2307/1968975
Jean Poncet, Groupes de Lie compacts de transformations de l’espace euclidien et les sphères comme espaces homogènes, Comment. Math. Helv. 33 (1959), 109–120 (French). MR 103946, DOI 10.1007/BF02565911
Hans Scheerer, Homotopieäquivalente kompakte Liesche Gruppen, Topology 7 (1968), 227–232 (German). MR 229259, DOI 10.1016/0040-9383(68)90003-7
Victor Schneider, Transitive actions on highly connected spaces, Proc. Amer. Math. Soc. 38 (1973), 179–185. MR 321125, DOI 10.1090/S0002-9939-1973-0321125-9
Emery Thomas, Exceptional Lie groups and Steenrod squares, Michigan Math. J. 11 (1964), 151–156. MR 163307