Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-21T04:48:01.727Z Has data issue: false hasContentIssue false
  • English
  • Français

SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

Published online by Cambridge University Press:  07 February 2017

DAISUKE KISHIMOTO  and
MITSUNOBU TSUTAYA
DAISUKE KISHIMOTO
Affiliation:
Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan e-mail: [email protected]
MITSUNOBU TSUTAYA
Affiliation:
Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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.

A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n(p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.

Keywords

55Q15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 165 - 174
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Adams, J. F., The sphere, considered as an H-space mod p , Quart. J. Math. 12 (1961), 52–60.Google Scholar
2. Bott, R., A note on the Samelson products in the classical groups, Comment. Math. Helv. 34 (1960), 249–256.Google Scholar
3. Friedlander, E. M., Exceptional isogenies and the classifying spaces of simple Lie groups, Ann. Math. 101 (1975), 510–520.Google Scholar
4. Hamanaka, H. and Kono, A., A note on the Samelson products in π*(SO(2n)) and the group [SO(2n), SO(2n)], Topology Appl. 154 (3) (2007), 567–572.Google Scholar
5. Hamanaka, H. and Kono, A., A note on Samelson products and mod p cohomology of classifying spaces of the exceptional Lie groups, Topology Appl. 157 (2) (2010), 393–400.Google Scholar
6. Hasui, S., Kishimoto, D. and Ohsita, A., Samelson products in p-regular exceptional Lie groups, Topology Appl. 178 (1) (2014), 17–29.Google Scholar
7. James, I. M., On homotopy theory of classical groups, Ann. Acad. Brasil. Cienc. 39 (1967), 39–44.Google Scholar
8. Kumpel, P. G., Mod p-equivalences of mod p H-spaces, Quart. J. Math. 23 (1972), 173–178.Google Scholar
9. Kaji, S. and Kishimoto, D., Homotopy nilpotency in p-regular loop spaces, Math. Z. 264 (1) (2010), 209–224.Google Scholar
10. Kono, A. and Oshima, H., Commutativity of the group of self homotopy classes of Lie groups, Bull. London Math. Soc. 36 (2004), 37–52.CrossRefGoogle Scholar
11. Lin, J., H-spaces with finiteness conditions, in Handbook of algebraic topology (James I. M., Editors) (Elsevier, North-Holland, 1995), 1095–1141, Chapter 22.Google Scholar
12. Mahowald, M., A Samelson product in SO(2n), Bol. Soc. Math. Mexicana 10 (1965), 80–83.Google Scholar
13. Morisugi, K., Hopf construction, Samelson products and suspension maps, Contemporary Math. 239 (1999), 225–238.Google Scholar
14. Toda, H., Composition methods in homotopy groups of spheres, Ann. of Math. Studies, vol. 49 (Princeton University Press, Princeton N.J., 1962).Google Scholar