Abstract
Actions of compact Lie groups on the homogeneous spaces G/NT, G a compact connected semisimple Lie group, NT⊂G the normalizor of a maximal torus T in G, are considered. If the acting group is a torus, the action is lifted to the universal covering G/T and the corresponding equivariant cohomology is computed for a coefficient field of characteristic O. The symmetry degree of all homogeneous spaces G/NT is computed confirming a conjecture of W. Y. Hsiang. The nonexistence of fixed points of semisimple compact Lie group actions on G/NT is proved in the case that the group acts differentiably and effectively.
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Literatur
Borel, A.: Seminar on Transformation Groups, Annals of Mathematics Studies 76, Princeton, Princeton Univ. Press, 1960
Bredon, G.: The cohomology ring structure of a fixed point set, Ann. of Math. 80, 524–537, 1964
Bredon, G.: Introduction to compact Transformation Groups, Academic Press, 1972, pp. 369–399
Chang, T.: On the number of relations in the cohomology of a fixed point set, Manuscripta Mathematica 18, 1976, p. 245
Gottlieb, H. D.: Lifting Group actions in fibrations In Geometric Applications of Homotopy Theory, Proceedings (Evanston 1977) pp. 217–254, Lecture Notes in Mathematics 57, Berlin-Heidelberg-New York, Springer 1978
Hattori, A., Yoshida T.: Lifting compact group actions in fiber bundles, Jap. J. Math. (N. S.) 2., 13–25 (1976)
Hauschild, V.: Aktionen kompakter Liegruppen mit linearer äquivarianter Kohomologie, Manuskript, Konstanz 1980
Hsiang, W. Y.: Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 85, Berlin-Heidelberg-New York, Springer 1975
Hsiang, W. C., Hsiang, W. Y.: Degree of symmetries of homotopy spheres, Ann. of Math. 89, 52–67 (1969)
Montgomery, D.: Orbits of highest dimension, Seminar on Transformation Groups, Annals of Mathematics Studies 76, Princeton, Princeton Univ. Press, 1960
Mc Leod, J.: The Künneth Formula in Equivariant K-Theory In “Symposion on Algebraic Topology”, Waterloo, 1978, Springer Lecture Notes 741
Puppe, V.: Cohomology of Fixed Point sets and Deformation of Algebras, Manuscripta Mathematica 23, 343–354 (1978)
Snaith, V.: On the Künneth formula spectral sequence in equivariant K-Theory, Proceedings Camb. Phil. Soc., 72 (1972), 167–177
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Hauschild, V. Über den Symmetriegrad der rational azyklischen kompakten homogenen Räume. Manuscripta Math 32, 365–379 (1980). https://doi.org/10.1007/BF01299610
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DOI: https://doi.org/10.1007/BF01299610