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Whitehead products in the complex Stiefel manifolds

Published online by Cambridge University Press:  20 January 2009

Yasukuni Furukawa
Yasukuni Furukawa
Affiliation:
Department of Mathematics, Aichi University of Education, Kariya, Japan 448
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The complex Stiefel manifold Wn,k, where n≦k≦1, is a space whose points are k-frames in Cn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 2 , June 1983 , pp. 241 - 251
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Hoo, C. S. and Mahowald, M. E., Some homotopy groups of Stiefel manifolds, Bull. Amer. Math. Soc. 71 (1965), 661–667.CrossRefGoogle Scholar
2.James, I. M., Note on Stiefel manifolds I, Bull. London Math. Soc. 2 (1970), 199–203.CrossRefGoogle Scholar
3.Kristensen, L. and Madsen, I., Note on Whitehead products in spheres, Math. Scand. 21(1967), 301–314.CrossRefGoogle Scholar
4.McCarty, G. S. Jr. Products between homotopy groups and the j-morphism, Quart. J. Math. Oxford (2) 15 (1964), 362–370.CrossRefGoogle Scholar
5.Nomura, Y., Some homotopy groups of real Stiefel manifolds in the metastable range I-V, Sci. Rep. Coll. Gen. Ed. Osaka Univ. 27 (1978), 1–31, 55–97; 28 (1979), 1–26, 35–60; 29 (1980),159–183.Google Scholar
6.Nomura, Y. and Furukawa, Y., Some homotopy groups of complex Stiefel manifolds, Sci. Rep. Coll. Gen. Ed. Osaka Univ. 25 (1976), 1–17.Google Scholar
7.Nomura, Y. and Furukawa, Y., Whitehead products in πr(X n·2) to appear.Google Scholar
8.Paechter, G. F., The groups πr(V n,m)I-v, Quart. J. Math. Oxford (2) 7 (1956), 249–268; 9(1958), 8–27; 10 (1959), 17–37, 241–260; ll (1960), 1–16.Google Scholar
9.Sigrist, F., Groupes d'homotopie des variétés de Stiefel complexes, Comment. Math. Helv. 43(1968), 121–131.CrossRefGoogle Scholar
10.Thomeier, S., Einige Ergebnisse über Homotopiegruppen von Sphären, Math. Ann. 164 (1966), 225–250.CrossRefGoogle Scholar