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Elementary abelian \(\varvec{p}\)-groups are the only finite groups with the Borsuk–Ulam property

  • Ikumitsu NagasakiEmail author
Article
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Abstract

It is well known that the Borsuk–Ulam theorem holds for elementary abelian p-groups \(C_p{}^k\). When the Borsuk–Ulam theorem holds for a finite group G, we say that G has the Borsuk–Ulam property or G is a BU-group. In this paper, we show that a non-abelian p-group of exponent p is not a BU-group, which leads to a complete classification of finite BU-groups, namely finite BU-groups are only elementary abelian p-groups.

Keywords

Borsuk–Ulam theorem Borsuk–Ulam property BU-group representation sphere equivariant map 

Mathematics Subject Classification

Primary 55M20 Secondary 57S17 

Notes

Acknowledgements

The author would like to thank the referee for carefully reading the manuscript and for giving valuable comments and suggestions.

References

  1. 1.
    Bartsch, T.: On the existence of Borsuk–Ulam theorems. Topology 31, 533–543 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Błaszczyk, Z., Marzantowicz, W., Singh, M.: Equivariant maps between representation spheres. Bull. Belg. Math. Soc. Simon Stevin 24, 621–630 (2017)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biasi, C., de Mattos, D.: A Borsuk-Ulam Theorem for compact Lie group actions. Bull. Braz. Math. Soc. 37, 127–137 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dold, A.: Simple proofs of some Borsuk–Ulam results. In: Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemp. Math., vol. 19, pp. 65–69 (1983)Google Scholar
  5. 5.
    Fadell, E., Husseini, S.: An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems. Ergod. Theory Dyn. Syst. 8, 73–85 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gorenstein, D.: Finite Groups, 2nd edn. AMS Chelsea, Providence (2007)zbMATHGoogle Scholar
  7. 7.
    Husemoller, D.: Fiber Bundles, Graduate Texts in Mathematics, vol. 20. Springer, Berlin (1993)Google Scholar
  8. 8.
    Kobayashi, T.: The Borsuk–Ulam theorem for a \({\mathbb{Z}}_q\)-map from a \({\mathbb{Z}}_q\)-space to \(S^{2n+1}\). Proc. Am. Math. Soc. 97, 714–716 (1986)Google Scholar
  9. 9.
    Komiya, K.: Equivariant \(K\)-theoretic Euler classes and maps of representation spheres. Osaka J. Math. 38, 239–249 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Leary, I.J.: The mod-\(p\) cohomology rings of some \(p\)-groups. Math. Proc. Camb. Philos. Soc. 112, 63–75 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lewis, G.: The integral cohomology rings of groups of order \(p^3\). Trans. Am. Math. Soc. 132, 501–529 (1968)zbMATHGoogle Scholar
  12. 12.
    Marzantowicz, W.: An almost classification of compact Lie groups with Borsuk–Ulam properties. Pac. J. Math. 144, 299–311 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marzantowicz, W.: Borsuk-Ulam theorem for any compact Lie group. J. Lond. Math. Soc. II. Ser. 49, 195–208 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Marzantowicz, W., de Mattos, D., dos Santos, E.L.: Bourgin–Yang versions of the Borsuk–Ulam theorem for \(p\)-toral groups. J. Fixed Point Theory Appl. 19, 1427–1437 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Matoušek, J.: Using the Borsuk–Ulam Theorem. Springer, Berlin (2003)zbMATHGoogle Scholar
  16. 16.
    Nagasaki, I., Kawakami, T., Hara, Y., Ushitaki, F.: The Smith homology and Borsuk–Ulam type theorems. Far East J. Math. Sci. 38, 205–216 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Serre, J.P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42. Springer, Berlin (1977)CrossRefGoogle Scholar
  18. 18.
    Steinlein, H.: Borsuk’s Antipodal Theorem and Its Generalizations and Applications: A Survey, Topological Methods in Nonlinear Analysis, pp. 166–235, Montreal (1985)Google Scholar
  19. 19.
    Steinlein, H.: Spheres and symmetry: Borsuk’s antipodal theorem. Topol. Methods Nonlinear Anal. 1, 15–33 (1993)MathSciNetCrossRefGoogle Scholar
  20. 20.
    tom Dieck, T.: Transformation Groups and Representation Theory, Lecture Note in Mathematics, vol. 766. Springer, Berlin (1979)Google Scholar
  21. 21.
    tom Dieck, T.: Transformation Groups. Walter de Gruyter, Berlin (1987)CrossRefGoogle Scholar
  22. 22.
    Waner, S.: A note on the existence of \(G\)-maps between spheres. Proc. Am. Math. Soc. 99, 179–181 (1987)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsKyoto Prefectural University of MedicineKyotoJapan

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