Skip to main content

Concise tables of James numbers and some homotopy of classical Lie groups and associated homogeneous spaces

Part of the Lecture Notes in Mathematics book series (LNM,volume 1509)

Keywords

  • Projective Space
  • Symmetric Space
  • Homogeneous Space
  • Homotopy Group
  • Complex Projective Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • M. Barratt and M. Mahowald, The metastable homotopy of O(n), Bull. Amer. Math Soc. 70 (1964), 758–760.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces II, Amer. J. Math. 81 (1959), 103–119.

    CrossRef  MathSciNet  Google Scholar 

  • R. Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959),313–337.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • R. Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960), 249–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • M. Crabb and K. Knapp, James numbers and the codegree of vector bundles I, II, (preprint).

    Google Scholar 

  • M. Crabb and K. Knapp, The Hurewicz map on stunted complex projective spaces, Amer. J. Math 110 (1988), 783–809.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • M. Crabb and K. Knapp, James numbers, Math. Ann. 282 (1988), 395–422.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • E. Curtis and M. Mahowald, The unstable Adams spectral sequence for the 3-sphere, (preprint).

    Google Scholar 

  • Y. Hirashima and H. Oshima, A note on stable James numbers of projective spaces, Osaka J. Math. 13 (1976), 157–161.

    MathSciNet  MATH  Google Scholar 

  • B. Harris, On the homotopy groups of the classical groups, Ann. of Math. 74 (1961),407–413.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • B. Harris, Some calculations of homotopy group of symmetric spaces, Trans. Amer. Math. Soc. 106 (1963), 174–184.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • C. Hoo and M. Mahowald, Some homotopy groups of Stiefel manifolds, Bull. Amer. Math. Soc. 71 (1965), 661–667.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • H. Imanishi, Unstable homotopy groups of classical groups (odd primary components), J. Math. Kyoto Univ. 7 (1968), 221–243.

    MathSciNet  MATH  Google Scholar 

  • M. Imaoka and K. Morisugi, On the stable Hurewicz image of some stunted projective spaces I, Pub. RIMS Kyoto Univ. 39 (1984), 839–852.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • M. Imaoka and K. Morisugi, On the stable Hurewicz image of some stunted projective spaces II, Pub. RIMS Kyoto Univ. 39 (1984), 853–866.

    CrossRef  MATH  Google Scholar 

  • M. Imaoka and K. Morisugi, On the stable Hurewicz image of some stunted projective spaces III, Mem. Fac. Sci. Kyushu Univ. 39 (1985), 197–208.

    MathSciNet  MATH  Google Scholar 

  • H. Kachi, Homotopy groups of the homogeneous space SU(n)/SO(n), J. Fac. Sci. Shinshu Univ. 13 (1978), 27–34.

    MathSciNet  MATH  Google Scholar 

  • H. Kachi, Homotopy groups of the homogeneous space Sp(n)/U(n), J. Fac. Sci. Shinshu Univ. 13 (1978), 36–41.

    MathSciNet  MATH  Google Scholar 

  • H. Kachi, Homotopy groups of symmetric spaces Γn, J. Fac. Sci. Shinshu Univ. 13 (1978), 103–120.

    MathSciNet  MATH  Google Scholar 

  • M. Kervaire, Some nonstable homotopy groups of Lie groups, Ill. J. Math. 4 (1960), 161–169.

    MathSciNet  MATH  Google Scholar 

  • K. Knapp, Some applications of K-theory to framed bordism: e-invariant and transfer, Habilitationsschrift, Bonn (1979).

    Google Scholar 

  • M. Mahowald, The metastable homotopy of S n, Mem. Amer. Math. Soc. 72 (1968).

    Google Scholar 

  • M. Mahowald, On the metastable homotopy of SO(n), Proc. Amer. Math. Soc. 19 (1968), 639–641.

    MathSciNet  MATH  Google Scholar 

  • H. Matsunaga, The homotopy groups π2n+i(U(n)) for i=3,4 and 5, Mem. Fac. Sci. Kyushu Univ. 15 (1961), 72–81.

    MathSciNet  MATH  Google Scholar 

  • H. Matsunaga, The groups π2n+7(U(n)), odd primary components, Mem. Fac. Sci. Kyushu 16 (1962), 66–74.

    MathSciNet  MATH  Google Scholar 

  • H. Matsunaga, Applications of functional cohomology operations to the calculus of π2n+i(U(n)) for i=6 and 7, n≥4, Mem. Fac. Sci. Kyushu Univ. 17 (1963), 29–62.

    MathSciNet  MATH  Google Scholar 

  • H. Matsunaga, Unstable homotopy groups of Unitary groups (odd primary components), Osaka J. Math. 1 (1964), 15–24.

    MathSciNet  MATH  Google Scholar 

  • M. Mimura, On the generalized Hopf construction and the higher composition II, J. Math. Kyoto Univ. 4 (1965), 301–326.

    MathSciNet  Google Scholar 

  • M. Mimura, Quelques groupes d'homotopie metastables des espaces symetriques Sp(n) et U(2n)/Sp(n), C.R. Acad. Sci. Paris, 262 (1966), 20–21.

    MathSciNet  MATH  Google Scholar 

  • M. Mimura, Homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ. 6 (1967), 131–176.

    MathSciNet  MATH  Google Scholar 

  • M. Mimura, M. Mori and N. Oda, On the homotopy groups of spheres, Proc. Japan Acad. 50 (1974), 277–280.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • M. Mimura and H. Toda, The (n+20)-th homotopy groups of n-spheres, J. Math. Kyoto Univ. 3 (1963), 37–58.

    MathSciNet  MATH  Google Scholar 

  • M. Mimura and H. Toda, Homotopy groups of SU(3),SU(4) and Sp(2), J. Math. Kyoto Univ. 3 (1964), 217–250.

    MathSciNet  MATH  Google Scholar 

  • M. Mimura and H. Toda, Homotopy groups of symplectic groups, J. Math. Kyoto Univ. 3 (1964), 251–273.

    MathSciNet  MATH  Google Scholar 

  • H. Minami, A remark on odd-primary components of special unitary groups, Osaka J. Math. 21 (1984), 457–460.

    MathSciNet  MATH  Google Scholar 

  • M. Mori, Applications of secondary e-invariants to unstable homotopy groups of spheres, Mem. Fac. Sci. Kyushu Univ. 29 (1974), 59–87.

    MathSciNet  MATH  Google Scholar 

  • K. Morisugi, Homotopy groups of symplectic groups and the quaternionic James numbers, Osaka J. Math. 23 (1986), 867–880.

    MathSciNet  MATH  Google Scholar 

  • K. Morisugi, Metastable homotopy groups of Sp(n), J. Math. Kyoto Univ. 27 (1987), 367–380.

    MathSciNet  MATH  Google Scholar 

  • K. Morisugi, On the homotopy group π4n+16(Sp(n)) for n≥ 4, Bull. Fac. Ed. Wakayama Univ. (1988), 19–23.

    Google Scholar 

  • K. Morisugi, On the homotopy group, π8n+4(Sp(n)) and the Hopf invariant, to appear in J. Math Kyoto Univ.

    Google Scholar 

  • R. Mosher, Some stable homotopy of complex projective space, Topology 7 (1969), 179–193.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • R. Mosher, Some homotopy of stunted complex projective space, Ill. J. Math. 13 (1969), 192–197.

    MathSciNet  MATH  Google Scholar 

  • J. Mukai, The S1-transfer map and homotopy groups of suspended complex projective spaces, Math. J. Okayama Univ. 24 (1982), 179–200.

    MathSciNet  MATH  Google Scholar 

  • N. Oda, On the 2-components of the unstable homotopy groups of spheres I, Proc. Japan Acad. 53 (1977), 202–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • N. Oda, On the 2-components of the unstable homotopy groups of shperes II, Proc. Japan Acad. 53 (1977), 215–218.

    CrossRef  MathSciNet  Google Scholar 

  • N. Oda, Some homotopy groups of SU(3),SU(4) and Sp(2), Fukuoka Univ. Sci. Reports 8 (1978), 77–90.

    MATH  Google Scholar 

  • N. Oda, Periodic families in the homotopy groups of SU(3),SU(4),Sp(2) and G2, Mem. Fac. Sci. Kyushu Univ. 32 (1978), 277–290.

    MathSciNet  MATH  Google Scholar 

  • S. Oka, On the homotopy groups of sphere bundles over spheres, J. Sci. Hiroshima Univ. 33 (1969), 161–195.

    MathSciNet  MATH  Google Scholar 

  • H. Ōshima, On the stable James numbers of complex projective spaces, Osaka J. Math. 11 (1974), 361–366.

    MathSciNet  MATH  Google Scholar 

  • H. Ōshima, On stable James numbers of stunted complex or quaternionic projective spaces, Osaka J. Math. 16 (1979), 479–504.

    MathSciNet  MATH  Google Scholar 

  • H. Ōshima, On the homotopy group π2n+9(U(n)) for n ≥ 6, Osaka J. Math. 17 (1980), 495–511.

    MathSciNet  MATH  Google Scholar 

  • H. Ōshima, Some James numbers of Stiefel manifolds, Proc. Camb. Phil. Soc. 92 (1982), 139–161.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • H. Ōshima, A homotopy group of the symmetric space SO(2n)/U(n), Osaka J. Math. 21 (1984), 473–475.

    MathSciNet  MATH  Google Scholar 

  • H. Ōshima, A remark on James numbers of Stiefel manifolds, Osaka J. Math. 21 (1984), 765–772.

    MathSciNet  MATH  Google Scholar 

  • F. Sigrist, Groupes d'homotopie des varietes de Stiefel complexes, Comment. Math. Helv. 43 (1968), 121–131.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • H. Toda, A topological proof of theorems of Bott and Borel-Hirzebruch for homotopy groups of unitary groups, Mem. Fac. Sci. Kyoto Univ. 32 (1959), 103–119.

    MathSciNet  MATH  Google Scholar 

  • H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Studies 49, Princeton University Press 1962.

    Google Scholar 

  • H. Toda, On homotopy groups of S3-bundles over spheres, J. Math. Kyoto Univ. 2 (1963), 193–207.

    MathSciNet  MATH  Google Scholar 

  • H. Toda, On iterated suspensions I, J. Math. Kyoto Univ. 5 (1965), 87–142.

    MathSciNet  MATH  Google Scholar 

  • H. Toda, On iterated suspensions III, J. Math. Kyoto Univ. 8 (1968), 101–130.

    MathSciNet  MATH  Google Scholar 

  • G. Walker, Estimates for the complex and quaternionic James numbers, Quart. J. Math. 32 (1981), 467–489.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • G. Walker, The James numbers bn, n−3 for n odd, (preprint), 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Lundell, A.T. (1992). Concise tables of James numbers and some homotopy of classical Lie groups and associated homogeneous spaces. In: Aguadé, J., Castellet, M., Cohen, F.R. (eds) Algebraic Topology Homotopy and Group Cohomology. Lecture Notes in Mathematics, vol 1509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087515

Download citation

  • DOI: https://doi.org/10.1007/BFb0087515

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55195-9

  • Online ISBN: 978-3-540-46772-4

  • eBook Packages: Springer Book Archive