Osaka Journal of Mathematics

Equivariant maps between representation spheres of cyclic ${p}$-groups

Ko Ohashi

Full-text: Open access


This paper deals with necessary conditions for the existence of equivariant maps between the unit spheres of unitary representations of a cyclic $p$-group $G$. T. Bartsch gave a necessary condition for some unitary representations of $G$ by using equivariant $K$-theory. We give two necessary conditions following Bartsch's approach. One is a generalization of Bartsch's result for any unitary representation of $G$ which does not contain the trivial representation. The other is a stronger necessary condition for some special cases.

Article information

Osaka J. Math. Volume 54, Number 4 (2017), 647-659.

First available in Project Euclid: 20 October 2017

Permanent link to this document

Zentralblatt MATH identifier

Primary: 55M35: Finite groups of transformations (including Smith theory) [See also 57S17]
Secondary: 19L64: Computations, geometric applications


Ohashi, Ko. Equivariant maps between representation spheres of cyclic ${p}$-groups. Osaka J. Math. 54 (2017), no. 4, 647--659.

Export citation


  • M.F. Atiyah and D.O. Tall: Group representations, $\lambda$-rings and the $J$-homomorphism, Topology 8 (1969), 253–297.
  • T. Bartsch: On the genus of representation spheres, Comment. Math. Helv. 65 (1990), 85–95.
  • K. Komiya: Equivariant $K$-theory and maps between representation spheres, Publ. Res. Inst. Math. Sci. 31 (1995), 725–730.
  • K. Komiya: Equivariant maps between representation spheres of a torus, Publ. Res. Inst. Math. Sci. 34 (1998), 271–276.
  • K. Komiya: Equivariant $K$-theoretic Euler classes and maps of representation spheres, Osaka J. Math. 38 (2001), 239–249.
  • A. Liulevicius: Borsuk-Ulam theorems and $K$-theory degrees of maps; in Algebraic topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math. 1051, Springer, Berlin, 1984, 610–619.
  • D.M. Meyer: ${\bf Z}/p$-equivariant maps between lens spaces and spheres, Math. Ann. 312 (1998), 197–214.
  • H.J. Munkholm and M. Nakaoka: The Borsuk-Ulam theorem and formal group laws, Osaka J. Math. 9 (1972), 337–349.
  • S. Stolz: The level of real projective spaces, Comment. Math. Helv. 64 (1989), 661–674.
  • J.W. Vick: An application of $K$-theory to equivariant maps, Bull. Amer. Math. Soc. 75 (1969), 1017–1019.