Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T13:06:14.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic Evaluation of Certain Characteristic Classes(1)

Published online by Cambridge University Press:  20 November 2018

Clark D. Jeffries
Clark D. Jeffries*
Affiliation:
University of British Columbia, Vancouver, British Columbia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We have an alternative proof of the following result of Kervaire [2]:

Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973, pp. 219 - 223
Copyright
Copyright © Canadian Mathematical Society 1973

Footnotes

(1)

Research supported by a postdoctoral fellowship from the National Research Council of Canada.

References

1. Atiyah, M. F., Bott, R., and Shapiro, A., Clifford modules, Topology, 3 (1964), 3–38.Google Scholar
2. Kervaire, M., On the Pontryagin classes of certain SO(n)-bundles over manifolds, Amer. Math J. 80 (1958), 632–638.Google Scholar
3. Kervaire, M., Some nonstable homotopy groups of Lie groups, Illinois Math J. 4 (1960), 161–169.Google Scholar
4. Steenrod, N., The topology of fibre bundles, Princeton Univ. Press, Princeton, N.J., 1951.Google Scholar
5. Toda, H., Saito, Y., and Yokota, I., Note on the generator of π7(SO(n)), Memoirs Coll. Sci., Univ. Kyoto, Ser. A, 30 (1957), 227–230.Google Scholar
6. Zvengrowski, P., Canonical vector fields on spheres, Comment. Math. Helv. 43 (1968), 341–347.Google Scholar