Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-30T08:02:53.335Z Has data issue: false hasContentIssue false
  • English
  • Français

The Group of Units in K-Theory Modulo an Odd Prime

Published online by Cambridge University Press:  20 November 2018

R. J. Steiner
R. J. Steiner*
Affiliation:
University of Western Ontario, London, Ontario
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.

1. Introduction. There are several multiplicative cohomology theories for which the group of units in the zeroth term is the zeroth term of another cohomology theory. Examples, due to Segal, May and others, are given by ordinary cohomology with rather general graded coefficients, real and complex K-theory with integral coefficients, and various bordism theories, also with integral coefficients [8, 7, 2, 5, IV]. The object of this paper is to show that complex K-theory modulo an odd prime p provides a counter-example.

To state the theorem precisely we recall the result of Araki and Toda that there is a unique anticommutative associative admissible multiplication in K*( ;Z/p) for p an odd prime [3, 3, 7, 10]; admissible is defined in [3] and means essentially that the reduction homomorphism K*( ) → K*( ; Z/p) preserves products.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 915 - 928
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Adams, J. F., Lectures on generalised cohomology, in Category theory, homology theory and their applications III, Lecture notes in mathematics 99 (Springer, Berlin, Heidelberg, New York, 1969), 1–138.Google Scholar
2. Adams, J. F. and Priddy, S. B., Uniqueness of BSO, Math. Proc. Camb. Phil. Soc. 80 (1976), 475–509.Google Scholar
3. Araki, S. and Toda, H., Multiplicative structures in mod q cohomology theories, Osaka J. Math. 2 (1965), 71-115 and. 3 (1966), 81–120.Google Scholar
4. Cohen, F. R., Lada, T. J. and May, J. P., The homology of iterated loop spaces, Lecture notes in mathematics 533 (Springer, Berlin, Heidelberg, New York, 1976).Google Scholar
5. May, J. P., Ex ring spaces and Ex ring spectra, Lecture notes in mathematics 577 (Springer, Berlin, Heidelberg, New York, 1977).Google Scholar
6. Milnor, J. W. and Moore, J. C., On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211–264.Google Scholar
7. Segal, G., Categories and cohomology theories, Topolog. 13 (1974), 293–312.Google Scholar
8. Segal, G., The multiplicative group of classical cohomology, Quart. J. Math. Oxford (2). 26 (1975), 289–293.Google Scholar