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On the codegree of negative multiples of the Hopf bundle

Published online by Cambridge University Press:  14 November 2011

M. C. Crabb  and
K. Knapp
M. C. Crabb
Affiliation:
Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TY, Scotland
K. Knapp
Affiliation:
Fachbereich Mathematik, Bergische Universität-GHS Wuppertal, D 5600 Wuppertal 1, West Germany
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Synopsis

Let H be the Hopf line bundle over the complex projective space of complex dimension k – 1. We determine the codegree of the virtual bundle –nH in the range l ≦ n ≦ k. This codegree has geometric significance as a stable James number.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 107 , Issue 1-2 , 1987 , pp. 87 - 107
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Atiyah, M. F.. Power operations in K-theory. Quart. J. of Math. (Oxford) 17 (1966), 165–193.CrossRefGoogle Scholar
2Atiyah, M. F. and Singer, I. M.. The index of elliptic operators I. Ann. of Math. 87 (1968), 484–530.CrossRefGoogle Scholar
3Atiyah, M. F. and Singer, I. M.. The index of elliptic operators III. Ann. of Math. 87 (1968), 546–604.CrossRefGoogle Scholar
4Becker, J. C. and Schultz, R. E.. Equivariant function spaces and stable homotopy theory I. Comment. Math. Helv. 49 (1974), 1–34.CrossRefGoogle Scholar
5Crabb, M. C..The Euler class, the Euler characteristic and obstruction theory for monomorphisms of vector bundles (D. Phil, thesis, University of Oxford, 1975).Google Scholar
6Crabb, M. C.. Z/2-Homotopy Theory (Cambridge: Cambridge University Press, 1980).CrossRefGoogle Scholar
7Crabb, M. C. and Knapp, K.. James quasi-periodicity for the codegree of vector bundles over complex projective spaces. J. London Math. Soc. (2) 35 (1987), 353–366.CrossRefGoogle Scholar
8Crabb, M. C. and Knapp, K.. James numbers (preprint, 1986).Google Scholar
9Crabb, M. C., Knapp, K. and Morisugi, K.. On the stable Hurewicz image of stunted quaternionic projective spaces. In Homotopy Theory and Related Topics, Advanced Studies in Pure Mathematics 9, pp. 319–334 (Tokyo: Kinokuniya, 1987).CrossRefGoogle Scholar
10Crabb, M. C. and Sutherland, W. A.. The space of sections of a sphere-bundle I. Proc. Edinburgh Math. Soc. 29 (1986), 383–403.CrossRefGoogle Scholar
11Dieck, T. torn. The Burnside Ring and Equivariant Stable Homotopy. (Unpublished lecture notes, University of Chicago, 1975).Google Scholar
12Hauschild, H.. Zerspaltung äquivarianter Homotopiemengen. Math. Ann. 230 (1977), 279–292.CrossRefGoogle Scholar
13James, I. M.. Spaces associated with Stiefel manifolds. Proc. London Math. Soc. 9 (1959), 115–140.CrossRefGoogle Scholar
14James, I. M.. The Topology of Stiefel Manifolds (Cambridge: Cambridge University Press, 1976).Google Scholar
15Knapp, K.. Some applications of K-theory to framed bordism. (Habilitationsschrift, Bonn, 1979).Google Scholar
16Knapp, K.. On the bi-stable J-homomorphism. In Algebraic Topology Aarhus 1978, Lecture Notes in Mathematics 763, pp. 13–22. (Berlin: Springer, 1979).Google Scholar
17Morisugi, K.. Stable Self Maps of the Quaternionic (Quasi-) Projective Space. Publ. RIMS. Kyoto Univ. 20 (1984), 971–976.CrossRefGoogle Scholar
18Mukai, J. and Oka, S.. A note on the quaternionic quasi-projective space. Mem. Faculty of Science, Kyushu Univ. A 38 (1984), 277–284.Google Scholar