Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T03:53:50.998Z Has data issue: false hasContentIssue false
  • English
  • Français

Bott Periodicity and the Parallelizability of the spheres

Published online by Cambridge University Press:  24 October 2008

M. F. Atiyah  and
F. Hirzebruch
M. F. Atiyah
Affiliation:
Pembroke College, Cambridge
F. Hirzebruch
Affiliation:
Mathematisches Institut der Universität, Bonn
Get access Rights & Permissions [Opens in a new window]

Extract

The theorems of Bott (4), (5) on the stable homotopy of the classical groups imply that the sphere Sn is not parallelizable for n ≠ 1, 3, 7. This was shown independently by Kervaire(8) and Milnor(7), (9). Another proof can be found in (3), §26·11. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that Sn with any (perhaps extraordinary) differentiable structure is not parallelizable if n ≠ 1, 3, 7. Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in (4), (5). The purpose of this note is to show how the refined form of Bott's results given in (6) leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the following theorem due to Milnor (9) which implies the non-parallelizability.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 2 , April 1961 , pp. 223 - 226
Copyright
Copyright © Cambridge Philosophical Society 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Atiyah, M. F., and Hirzebruch, F., Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Soc. 65 (1959), 267–81.CrossRefGoogle Scholar
(2)Atiyah, M. F., and Hirzebruch, F., Vector bundles and homogeneous spaces. Symposium on Differential Geometry, Tucson, Arizona, February 1960 (to appear).CrossRefGoogle Scholar
(3)Borel, A., and Hirzebruch, F., Characteristic classes and homogeneous spaces I, II. Amer. J. Math. 80 (1958), 458–538; 81 (1959), 315–82.CrossRefGoogle Scholar
(4)Bott, R., The space of loops on a Lie group. Mich. Math. J. 5 (1958), 35–61.CrossRefGoogle Scholar
(5)Bott, R., The stable homotopy of the classical groups. Ann. Math. 70 (1959), 313–37.CrossRefGoogle Scholar
(6)Bott, R., Some remarks on the periodicity theorems. Collogue de Topologie, Lille, 1959 Bull. Soc. math. Fr. 87 (1959), 293–310.Google Scholar
(7)Bott, R., and Milnor, J., On the parallelizability of the spheres. Bull. Amer. Math. Soc. 64 (1958), 87–9.CrossRefGoogle Scholar
(8)Kervaire, M. A., Non-parallelizability of the n-sphere for n > 7. Proc. Nat. Acad. Sci., Wash., 44 (1958), 280–3.CrossRefGoogle ScholarPubMed
(9)Milnor, J., Some consequences of a theorem of Bott. Ann. Math. 68 (1958), 444–9.CrossRefGoogle Scholar