Journal of the Mathematical Society of Japan

On framed simple Lie groups

Haruo MINAMI

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a compact simple Lie group $G$, we show that the element $[G, \mathcal{L}] \in \pi^S_*(S^0)$ represented by the pair $(G, \mathcal{L})$ is zero, where $\mathcal{L}$ denotes the left invariant framing of $G$. The proof relies on the method of E. Ossa [Topology, 21 (1982), 315–323].

Article information

Source
J. Math. Soc. Japan Volume 68, Number 3 (2016), 1219-1229.

Dates
First available in Project Euclid: 19 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956166

Digital Object Identifier
doi:10.2969/jmsj/06831219

Mathematical Reviews number (MathSciNet)
MR3523545

Zentralblatt MATH identifier
1351.57029

Subjects
Primary: 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Secondary: 22E46: Semisimple Lie groups and their representations 19L20: $J$-homomorphism, Adams operations [See also 55Q50]

Keywords
framed manifolds Lie groups Adams conjecture

Citation

MINAMI, Haruo. On framed simple Lie groups. J. Math. Soc. Japan 68 (2016), no. 3, 1219--1229. doi:10.2969/jmsj/06831219. https://projecteuclid.org/euclid.jmsj/1468956166.


Export citation

References

  • J. F. Adams, Lectures on Exceptional Lie Groups, Chicago Lectures in Math., University of Chicago Press, Chicago, 1996.
  • M. F. Atiyah, Power operations in $K$-theory, Quart. J. Math. Oxford Ser. (2), 17 (1966), 165–193.
  • M. F. Atiyah and L. Smith, Compact Lie groups and the stable homotopy of spheres, Topology, 13 (1974), 135–142.
  • J. C. Becker and R. E. Schultz, Fixed point indices and left invariant framings, In Geometric Applications of Homotopy Theory I, Proc. Evanston, 1977, Springer Lecture Notes, 657, 1–31.
  • H. H. Gershenson, A problem in compact Lie groups and cobordism, Pac. J. of Math., 51 (1974), 121–129.
  • K. Knapp, Rank and Adams filtration of a Lie group, Topology, 17 (1978), 41–52.
  • E. Ossa, Lie groups as framed manifolds, Topology, 21 (1982), 315–323.
  • H. Sati, M-Theory with Framed Corners and Tertiary Index Invariants, SIGMA, 10 (2014), 024 arXiv:1203.4179 [hep-th].
  • H. U. Schön, Lie-Gruppen in der Novikov-Spektralsequenz, Ph. D. Dissertation, Univ. Wuppertal, 1983.
  • B. Steer, Orbits and the homotopy class of a compactification of a classical map, Topology, 15 (1976), 383–393.
  • H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Studies, 49, Princeton University Press, 1962.
  • R. M. W. Wood, Framing the exceptional Lie group $G_2$, Topology, 15 (1976), 303–320.