Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:53:46.500Z Has data issue: false hasContentIssue false
  • English
  • Français

A Descent Theorem for Hermitian K-Theory

Published online by Cambridge University Press:  20 November 2018

Victor Snaith
Victor Snaith*
Affiliation:
The 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.

Let KO and KU respectively denote the real and complex periodic K-theory spectra [1, Part III]. Let KSC denote the spectrum representing self-conjugate K-theory [2, G]. Thus we have a fibring

1.1

where T is induced by complex conjugation on the unitary group.

The following result is due to R. Wood [1, p. 206] and, I believe, to D. W. Anderson.

1.2. PROPOSITION. Let generate the stable one-stem. Then there are weak equivalences of spectra

a

and

b

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 835 - 847
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Adams, J. F., Stable homotopy and generalised homology, Chicago Lecture Notes in Mathematics, (1974).Google Scholar
2. Anderson, D. W., The real K-theory of classifying spaces, Proc. Nat. Acad. Sci. 57 (1964), 634–636.Google Scholar
3. Atiyah, M. F., K-theory and reality, Quart. J. Math., Oxford (2), 17 (1966), 367–386.Google Scholar
4. Browder, W., Algebraic K-theory with coefficients Z/p, Springer Verlag Lecture Notes in Mathematics 654 (1978), 40–84.Google Scholar
5. Delzant, M. A., Definition des classes de Stiefel-Whitney d'un module quadratique sur un corps de charactéristique différente de 2, C.R. Acad. Sci., Paris 255 (1962), 1366–1368.Google Scholar
6. Gabber, O., Lecture at France — U.S.A. K-theory conference, Luminy (1983).Google Scholar
7. Gillet, H. and Thomason, R. W., The K-theory of strict Hensel local rings and a theorem of Suslin, JPAA 34 (1984), 241–251.Google Scholar
8. Grayson, D. (after D. G. Quillen), Higher algebraic K-theory II, Springer-Verlag Lecture Notes in Mathematics 551, 217–240.CrossRefGoogle Scholar
9. Green, P. S., A cohomology theory based upon selfconjugacies of complex vector bundles. Bull. A.M. Soc. (1964), 522–524.Google Scholar
10. Jardine, J. F., Simplicial objects in a Grothendieck topos, preprint (1983).Google Scholar
11. Jardine, J. F., A rigidity theorem for L-theory, preprint (1983).Google Scholar
12. Karoubi, M., Théorie de Quillen et homologie du groupe orthogonal, Ann. Math. 112 (1980), 206–257.Google Scholar
13. Karoubi, M., Le théorème fondamental de la K-theorie hermitienne, Ann. Math. 112 (1980), 259–282.Google Scholar
14. Karoubi, M., Homology of the infinite orthogonal and symplectic groups over algebraically closed fields, Inventiones Math. 73 (1983), 247–250.Google Scholar
15. Karoubi, M., Relations between algebraic K-theory and Hermitian K-theory, preprint (1984).CrossRefGoogle Scholar
16. Kahn, B., La deuxième classe de Stiefel-Whitney d'une représentation régulière, I & II, C.R. Acad. Sci., Paris 297 (1983), 313–316 and 573–576.Google Scholar
17. Kahn, B., Classes de Stiefel-Whitney de formes quadratiques et de représentations Galoisiennes reélles, Inventiones Math. 78 (1984), 223–256.Google Scholar
18. Milne, J. S., Étale cohomology, Princeton Math. Series 33 (1980).Google Scholar
19. Quillen, D. G., Higher algebraic K-theory I, Springer-Verlag Lecture Notes in Mathematics 341 (1973), 85–147.CrossRefGoogle Scholar
20. Serre, J-P., Sur 1-invariant de Witt de la forme Tr(x2), Comm. Math. Helv. 59 (1984), 651–676.Google Scholar
21. Snaith, V. P., Algebraic cobordism and K-theory, Mem. A. M. Soc. 227 (1979).Google Scholar
22. Snaith, V. P., Localised stable homotopy and algebraic K-theory, Mem. A. M. Soc. 280 (1983).Google Scholar
23. Snaith, V. P., A brief survey of Bott periodic K-theory, Can. Math, Soc. Conf. Proc. 2, Part I (1982).Google Scholar
24. Snaith, V. P., Stiefel-Whitney classes of symmetric bilinear forms — a formula of Serre, Can. Bull. Math. 28 (1985), 218–222.Google Scholar
25. Snaith, V. P., Algebraic K-theory and bilinear forms, in preparation.Google Scholar
26. Snaith, V. P., K-theory of the classifying spaces of Galois groups, to appear Proc. Conf., St. John's, Newfoundland (1983) in A. M. Soc. Contemporary Math, series.Google Scholar
27. Suslin, A. A., On the K-theory of algebraically closed fields, Inventiones Math. 73 (1983), 241–245.Google Scholar
28. Suslin, A. A., On the K-theory of local fields, to appear J. Pure and Appl. Alg.CrossRefGoogle Scholar
29. Thomason, R. W., Algebraic K-theory and étale cohomology, preprint.CrossRefGoogle Scholar
30. Wagoner, J. B., Delooping the classifying spaces of algebraic K-theory, Topology 11 (1972), 349–370.Google Scholar