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On Localized Self-Homotopy Groups of SU(4) and Sp(4)

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Abstract

In this article, first, localizing at odd primes p, we study the self-homotopy groups of SU(4) and Sp(4). Second, we consider the homotopy nilpotency of Sp(4) when localized at odd prime \(p \ge 7\).

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References

  1. Arkowitz, M.A., Oshima, H., Strom, J.: Noncommutativity of the group of self-homotopy classes of Lie groups. Topol. Appl. 25(1), 87–96 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Hamanaka, H.: On \([X, U(n)]\) when dim \(X\) is \(2n+1\). J. Math. Kyoto Univ. 44(3), 655–667 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Hamanaka, H.: Nilpotency of unstable \(K\)-theory. Topol. Appl. 154, 1368–1376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hamanaka, H., Kono, A.: On \([X, U(n)]\) when dim \(X = 2n\). J. Math. Kyoto Univ. 43(2), 333–348 (2003)

    MathSciNet  MATH  Google Scholar 

  7. James, I., Whitehead, J.: The homotopy theory of Sphere-bundles over spheres I. Proc. Lond. Math. Soc. 4, 196–218 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kishimoto, D.: Homotopy nilpotency in localized \( SU(n)\). Homol. Homotopy Appl. 11(1), 61–79 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kaji, S., Kishimoto, D.: Homotopy nilpotency in \(p\)-compact groups. Math. Z. 264(1), 209–224 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kishimoto, D., Kono, A., Tsutaya, M.: On localized unstable \(K^1\)-groups and applications to self-homotopy groups. Can. Math. Bull. 57(2), 344–356 (2014)

    Article  MATH  Google Scholar 

  11. Kono, A., Oshima, H.: Commutativity of the group of self-homotopy classes of Lie groups. Bull. Lond. Math. Soc. 36, 37–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mimura, M., Nishida, G., Toda, H.: Mod \(p\) decomposition of compact Lie groups. Publ. Res. Inst. Math. Sci. 13, 627–680 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mimura, M., Oshima, H.: Self-homotopy groups of Hopf spaces with at most three cells. J. Math. Soc. Jpn. 51(1), 71–92 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nagao, T.: On the groups \([X, Sp(n)]\) with dim \(X \le 4n+2\). J. Math. Kyoto Univ. 48–1, 149–165 (2008)

    MATH  Google Scholar 

  15. Oshima, H., Yagita, N.: Non-commutativity of self-homotopy groups. Kodai Math. J. 24, 15–25 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Toda, H.: Composition Methods in Homotopy Groups of Spheres, Annals of Mathematics Studies, vol. 49. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  17. Whitehead, G.W.: On mappings into group-like spaces. Comment. Math. Helv. 28, 320–328 (1954)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported (no.10/595) by Urmia University. Also, the author would like to thank the referee for a careful reading of the paper and for his helpful comments.

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Correspondence to Sajjad Mohammadi.

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Communicated by Mohammad Reza Koushesh.

In memory of Professor Mohammad Ali Asadi-Golmankhaneh.

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Mohammadi, S., Janfada, A.S. On Localized Self-Homotopy Groups of SU(4) and Sp(4). Bull. Iran. Math. Soc. 49, 69 (2023). https://doi.org/10.1007/s41980-023-00819-y

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  • DOI: https://doi.org/10.1007/s41980-023-00819-y

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