Advertisement

Acta Mathematica Sinica, English Series

, Volume 34, Issue 10, pp 1485–1500 | Cite as

Estimates of the Isovariant Borsuk–Ulam Constants of Connected Compact Lie Groups

  • Ikumitsu Nagasaki
Article
  • 11 Downloads

Abstract

The isovariant Borsuk–Ulam constant cG of a compact Lie group G is defined to be the supremum of c such that the inequality
$$c\left( {\dim V - \dim {V^C}} \right) \leqslant \dim W - \dim {W^G}$$
holds whenever there exists a G-isovariant map f: S(V) → S(W) between G-representation spheres. In this paper, we shall discuss some properties of cG and provide lower estimates of cG of connected compact Lie groups, which leads us to some Borsuk–Ulam type results for isovariant maps. We also introduce and discuss the generalized isovariant Borsuk–Ulam constant G for more general smooth G-actions on spheres. The result is considerably different from the case of linear actions.

Keywords

Isovariant map Borsuk–Ulam type theorem Borsuk–Ulam constant transformation groups representation theory 

MR(2010) Subject Classification

57S15 57S25 55M20 22E99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bartsch, T.: On the existence of Borsuk–Ulam theorems. Topology, 31, 533–543 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bartsch, T.: Topological methods for variational problems with symmetries. Lecture Notes in Math. 1560, Springer, Berlin Heidelberg, 1993Google 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bourbaki, N.: Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4–6, Chapters 7–9, Springer, New York, 2008zbMATHGoogle Scholar
  5. [5]
    Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups, Graduate Texts in Math. 98, Springer, New York, 1985CrossRefzbMATHGoogle Scholar
  6. [6]
    Bredon, G. E.: Introduction to Compact Transformation Groups, Academic Press, New York, 1972zbMATHGoogle Scholar
  7. [7]
    Dula, G., Schultz, R.: Diagram cohomology and isovariant homotopy theory. Mem. Amer. Math. Soc., 110, 1994Google Scholar
  8. [8]
    Hauschild, H.: Äquivariante Whiteheadtorsion. Manuscript. Math., 26, 63–82 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math. 9, Springer, Berlin-Heidelberg-New York, 1972CrossRefGoogle Scholar
  10. [10]
    Nagasaki, I.: The weak isovariant Borsuk–Ulam theorem for compact Lie groups. Arch. Math., 81, 348–359 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Nagasaki, I., Ushitaki, F.: New examples of the Borsuk–Ulam groups. RIMS Kôkyuroku Bessatsu, B39, 109–119 (2013)MathSciNetzbMATHGoogle Scholar
  12. [12]
    Nagasaki, I., Ushitaki, F.: A Hopf type classification theorem for isovariant maps from free G-manifolds to representation spheres. Acta Math. Sin., Engl. Ser., 27, 685–700 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Nagasaki, I., Kawakami, T., Hara, Y., et al.: The Smith homology and Borsuk–Ulam type theorems. Far East J. Math. Sci., 38, 205–216 (2010)MathSciNetzbMATHGoogle Scholar
  14. [14]
    Oliver, R.: Fixed-point sets of group actions on finite acyclic complexes. Comment. Math. Helv., 50, 155–177 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Oliver, R.: Smooth compact Lie group actions on disks. Math. Z., 149, 79–96 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Palais, R. S.: Classification of G-spaces. Mem. Amer. Math. Soc., 36, 1960Google Scholar
  17. [17]
    Schultz, R.: Isovariant homotopy equivalences of manifolds with group actions. Proc. Amer. Math. Soc., 144, 1363–1370 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Wasserman, A. G.: Isovariant maps and the Borsuk–Ulam theorem. Topolog. Appl., 38, 155–161 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKyoto Prefectural University of MedicineKyotoJapan

Personalised recommendations