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On the homotopy-symmetry of sphere bundles

Published online by Cambridge University Press:  24 October 2008

I. M. James
I. M. James
Affiliation:
Mathematical Institute, Oxford
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Extract

1. Introduction: Let B be a space and let E be an orthogonal (m + 1)-sphere bundle over B with projection p: E → B. Consider the fibre-preserving map T: E → E which is given by the antipodal transformation in each of the fibres. Following (4) we say that E is homotopy-symmetric if there exists a fibre-preserving homotopy of T into the identity map. Since T maps the fibres with degree (– 1)m this condition requires m to be even.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 2 , March 1971 , pp. 291 - 294
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Adams, J. F.On the non-existence of elements with Hopf invariant one. Ann. of Math. 72 (1960), 20–104.CrossRefGoogle Scholar
(2)Borel, A.Sur la cohomologie des espaces fibrés principaux…. Ann. of Math. 57 (1953), 115–207.CrossRefGoogle Scholar
(3)Eckmann, B.Stetige Lösungen linearer Gleichungssysteme. Comment. Math. Helv. 15 (1943), 318–339.CrossRefGoogle Scholar
(4)James, I. M.Note on sphere-bundles. I. Bull. American Math. Soc. 75 (1969), 617–621.CrossRefGoogle Scholar
(5)James, I. M.Note on Stiefel manifolds. I. Bull. London Math. Soc. 2 (1970), 199–203.CrossRefGoogle Scholar
(6)James, I. M. Note on Stiefel manifolds. II (to appear).Google Scholar
(7)Serre, J.-P.Groupes d'homotopie et classes de groupes abeliens. Ann. of Math. 58 (1953), 258–294.CrossRefGoogle Scholar
(8)Whitehead, O. W.Note on cross-sections in Stiefel manifolds. Comment. Math. Helv. 37 (1963), 239–240.Google Scholar