Annals of Global Analysis and Geometry

, Volume 6, Issue 2, pp 119–140 | Cite as

On a class of compact homogeneous spaces I

  • Alexander Shchetinin

Le K be a compact connected Lie group, L be a connected closed subgroup of K. It is well known that L is a subgroup of maximal rank of K if and only if the Euler characteristic of the manifold M = K/L is positive. Such homogeneous spaces M have been classified in [7, 10]. However, their topological classification was unknown. This classification is obtained in the present article. We show tha two compact homogeneous spaces M = K/L and M′ = K′/L′ of positive Euler characteristic are diffeomorphic if and only if the graded rings H*(M,Z) and H*(M′,Z) are isomorphic. We also obtain the rational homotopy classification of such homogeneous spaces which is not equivalent to the differential one. These results were announced in [15].


Present Article Group Theory Homogeneous Space Euler Characteristic Closed Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ahiezer, D. N.: Irreducible root systems and indecomposable homogeneous spaces (Russian). Teor. Funkcii, Funkcional. Anal. i Prilož., Har'kov 27 (1977), 22–26.Google Scholar
  2. [2]
    Bernstein, I. N., Gelfand, I. M., and Gelfand, S. I.: Shubert cells and cohomology of spaces G/P (Russian). Usp. Mat. Nauk 28: 3 (1973), 3–26.Google Scholar
  3. [3]
    Body, R., and Douglas, R.: Tensor products of graded algebras and unique factorisation. Amer. J. Math. 101: 4 (1979), 909–914.Google Scholar
  4. [4]
    Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groups de Lie compact. Ann. Math. 57 (1953), 115–207.Google Scholar
  5. [5]
    Borel, A.: On Kählerian coset spaces of semisimple Lie groups. Proc. Nat. Acad. Sci. USA 40 (1954), 1147–1154.Google Scholar
  6. [6]
    Borel, A., and Hirzebruch, F.: Characteristics classes and homogeneous spaces I. Amer. J. Math. 80: 2 (1958), 458–536.Google Scholar
  7. [7]
    Borel, A., and De Siebental, J.: Les sous-groupes fermés de rang maximum de groupes de Lie clos. Comment. Math. Helv. 23: 3 (1949), 200–221.Google Scholar
  8. [8]
    Bourbaki, N.: Groupes et algèbres de Lie, Chapitres 4–6, Eléments de Mathématique XXXIV, Paris (1968).Google Scholar
  9. [9]
    Doan Kuyng: Poincaré polynomials of compact Riemannian homogeneous spaces with irreducible stationary subgroup (Russian) Trudy Sem. Vect. Tensor Anal. 14 (1968), 33–93.Google Scholar
  10. [10]
    Dynkin, E. B.: Semisimple subalgebras of semisimple Lie algebras (Russian). Mat. Sb. (N. S.) 30(72): 4 (1952), 349–462.Google Scholar
  11. [11]
    Lemann, D.: Théorie homotopique des formes differentielle (d'après D. Sullivan). Société mathématique de France, Asterisque 45 (1977).Google Scholar
  12. [12]
    Onishchik, A. L.: Inclusion relation between transitive compact transformation groups (Russian). Trudy Moskov. Mat. Obshch. 11 (1962), 199–242.Google Scholar
  13. [13]
    Onishchik, A. L.: Transitive compact transformation groups (Russian). Mat. Sb. (N. S.) 60(102): 4 (1963), 447–485.Google Scholar
  14. [14]
    Onishchik, A. L.: Lie groups which are transitive on compact manifolds. III (Russian). Mat. Sb. 75: 2 (1968), 255–263.Google Scholar
  15. [15]
    Shchetinin, A. N.: Compact homogeneous spaces of positive Euler characteristic (Russian). Usp. Mat. Nauk 39: 2 (1984), 209–210.Google Scholar
  16. [16]
    Switzer, R.: Algebraic topology — homotopy and homology. Springer (1975).Google Scholar

Copyright information

© VEB Deutscher Verlag der Wissenschaften 1988

Authors and Affiliations

  • Alexander Shchetinin
    • 1
  1. 1.Leningradskii prospektMoskovskii avtomobil'no-doroznyi institytMoskvaUdSSR

Personalised recommendations