Skip to main content
×
×
Home

Self-Maps of Low Rank Lie Groups at Odd Primes

  • Jelena Grbić (a1) and Stephen Theriault (a2)
Abstract

Let G be a simple, compact, simply-connected Lie group localized at an odd prime p. We study the group of homotopy classes of self-maps [G, G] when the rank of G is low and in certain cases describe the set of homotopy classes ofmultiplicative self-maps H[G, G]. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Self-Maps of Low Rank Lie Groups at Odd Primes
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Self-Maps of Low Rank Lie Groups at Odd Primes
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Self-Maps of Low Rank Lie Groups at Odd Primes
      Available formats
      ×
Copyright
References
Hide All
[CN] Cohen, F. R. and Neisendorfer, J. A., A construction of p-local H-spaces. In: Algebraic Topology. Lecture Notes in Math. 1051. Springer, Berlin, 1984, pp. 351–359.
[CHZ] Cooke, G., Harper, J., and Zabrodsky, A., Torsion free mod p H-spaces of low rank. Topology 18(1979), no. 4, 349–359. doi:10.1016/0040-9383(79)90025-9
[Gra] Gray, B., EHP spectra and periodicity. I. Geometric constructions. Trans. Amer. Math. Soc. 340(1993), no. 2, 595–616. doi:10.2307/2154668
[Grb1] Grbić, J., Universal homotopy associative, homotopy commutative H-spaces and the EHP spectral sequence. Math. Proc. Cambridge Philos. Soc. 140(2006), no. 3, 377–400. doi:10.1017/S0305004106009182
[Grb2] Grbić, J., Universal spaces of two-cell complexes and their exponent bounds. Q. J. Math. 57(2006), no. 3, 355–366.
[GTW] Grbić, J., Theriault, S., and Wu, J., Suspension splittings and Hopf retracts of the loops on co-H spaces. http://www.math.nus.edu.sg/»matwujie/GTW.pdf
[H] Harris, B., On the homotopy groups of the classical groups. Ann. of Math. 74(1961), 407–413. doi:10.2307/1970240
[J1] James, I. M., Reduced product spaces. Ann. of Math. 62(1955), 170–197. doi:10.2307/2007107
[J2] James, I. M., On H-spaces and their homotopy groups. Quart. J. Math. Oxford 11(1960), 161–179. doi:10.1093/qmath/11.1.161
[Mc] Mc Gibbon, C. A., Homotopy commutativity in localized groups. Amer. J. Math 106(1984), no. 3, 665–687. doi:10.2307/2374290
[Mil] Miller, H. R., Stable splittings of Stiefel manifolds. Topology 24(1985), no. 4, 411–419. doi:10.1016/0040-9383(85)90012-6
[Mim] Mimura, M., The homotopy groups of Lie groups of low rank. J. Math. Kyoto Univ. 6(1967), 131–176.
[MNT1] Mimura, M., Nishida, G., and Toda, H., Localization of CW-complexes and its applications. J. Math. Soc. Japan 23(1971), 593–624.
[MNT2] Mimura, M., Mod p decomposition of compact Lie groups. Publ. Res. Inst. Math. Sci. 13(1977), no. 3, 627–680. doi:10.2977/prims/1195189602
[MO] Mimura, M. and Oshima, H., Self homotopy groups of Hopf spaces with at most three cells. J. Math. Soc. Japan 51(1999), no. 1, 71–92. doi:10.2969/jmsj/05110071
[MT] Mimura, M. and Toda, H., Cohomology operations and the homotopy of compact Lie groups. I. Topology 9(1970), 317–336. doi:10.1016/0040-9383(70)90056-X
[NY] Nishida, G. and Yang, Y.-M., On a p-local stable splitting of U(n). J. Math. Kyoto Univ. 41(2001), no. 2, 387–401.
[Th1] Theriault, S. D., The H-structure of low rank torsion free H-spaces. Q. J. Math. 56(2005), no. 3, 403–415. doi:10.1093/qmath/hah050
[Th2] Theriault, S. D., The odd primary H-structure of low rank Lie groups and its application to exponents. Trans. Amer. Math. Soc. 359(2007), no. 9, 4511–4535 (electronic). doi:10.1090/S0002-9947-07-04304-8
[To] Toda, H., On iterated suspensions. I. J. Math. Kyoto Univ. 5(1966), 87–142.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed