Numbers pp 281-302 | Cite as

Division Algebras and Topology

  • F. Hirzebruch
Part of the Graduate Texts in Mathematics book series (GTM, volume 123)

Abstract

The preceding chapters examined the division algebras of the real numbers, the complex numbers, the quaternions and the octonions. These are of dimension 1, 2, 4 and 8, respectively. So far no algebraist has been able to show that every division algebra has to be of one of these four dimensions, though this surprising fact can be proved by topological methods. Hopf was able to prove in 1940 [7], that the dimension of a division algebra must be a power of 2. His proof, which used the homology groups of projective spaces, will be given in §1. In the year 1958, Kervaire and Milnor independently of one another proved that the power of 2 must be equal to 1, 2, 4 or 8 [9]. They used for this purpose the periodicity theorem of Bott on the homotopy groups of unitary and orthogonal groups. The periodicity theorem had led to the development of K-theory ([4], [3]), a new cohomology theory with whose help many of the classical problems of topology, which had resisted the ordinary homology and cohomology theory, could be solved. We shall describe in §2 a proof of the (1, 2, 4, 8)-Theorem, which is based on K-theory.

Keywords

Manifold Dinates Paral Univer Betti 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. of Math. 72, 20–104 (1960).MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Adams, J.F.: Vector fields on spheres. Ann. of Math. 75, 603–632 (1962).MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Atiyah, M.F.: K-Theory. W.A. Benjamin, Inc., New York, Amsterdam 1967.Google Scholar
  4. [4]
    Atiyah, M.F., Hirzebruch, F.: Vector bundles and homogeneous spaces. Proc. of Symposia in Pure Mathematics, Vol. 3, pp. 7–38. Am. Math. Soc. 1961.MathSciNetGoogle Scholar
  5. [5]
    Atiyah, M.F., Hirzebruch, F.: Bott periodicity and the paral-lelizability of the spheres. Proc. Cambridge Phil. Soc. 57, 223–226 (1961).MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Hopf, H.: Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension. Fundamenta Math. 25, 427–440 (1935).Google Scholar
  7. [7]
    Hopf, H.: Ein topologischer Beitrag zur reellen Algebra. Comm. Math. Helvetici 13, 219–239 (1940/41).CrossRefGoogle Scholar
  8. [8]
    Hopf, H.: Einige persönliche Erinnerungen aus der Vorgeschichte der heutigen Topologie. Colloque de Topologie, Centre Beige de Recher-ches Mathématiques 1966, pp. 9–20.Google Scholar
  9. [9]
    Milnor, J.: Some consequences of a theorem of Bott. Ann. of Math. 68, 444–449 (1958).MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Milnor, J.: Topology from the differentiable viewpoint. The University Press of Virginia, 1965.Google Scholar
  11. [11]
    Milnor, J., Stasheff, J.: Characteristic classes. Ann. of Math. Studies 76, Princeton University Press 1974.Google Scholar
  12. [12]
    Samelson, H.: Heinz Hopf zum Gedenken. II. Zum wissenschaftlichen Werk von Heinz Hopf. Jber. Deutsch. Math., Verein 78, 126–146 (1976).MathSciNetGoogle Scholar
  13. [13]
    Steenrod, N.: The topology of fibre bundles. Princeton University Press 1951.Google Scholar
  14. [14]
    Stiefel, E.: Richtungsfelder und Fernparallelismus in n-dimen-sionalen Mannigfaltigkeiten. Comm. Math. Helvetici 8, 305–353 (1935/36).MathSciNetCrossRefGoogle Scholar
  15. [15]
    Stiefel, E.: Über Richtungsfelder in den projektiven Räumen und einen Satz aus der reellen Algebra. Comm. Math. Helvetici 13, 201–218 (1940/41).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York  1991

Authors and Affiliations

  • F. Hirzebruch

There are no affiliations available

Personalised recommendations