Topological Methods in Nonlinear Analysis

The Borsuk-Ulam property for cyclic groups

Marek Izydorek and Wacław Marzantowicz

Full-text: Open access


An orthogonal representation $V$ of a group $G$ is said to have the Borsuk-Ulam property if the existence of an equivariant map $f:S(W) \rightarrow S(V)$ from a sphere of representation $W$ into a sphere of representation $V$ implies that $\dim W \leq \dim V$. It is known that a sufficient condition for $V$ to have the Borsuk-Ulam property is the nontriviality of its Euler class ${\text{\bf e}}(V)\in H^{*} (BG;\mathcal R)$. Our purpose is to show that ${\text{\bf e}}(V) \neq 0 $ is also necessary if $G$ is a cyclic group of odd and double odd order. For a finite group $G$ with periodic cohomology an estimate for $G$-category of a $G$-space $X$ is also derived.

Article information

Topol. Methods Nonlinear Anal., Volume 16, Number 1 (2000), 65-72.

First available in Project Euclid: 22 August 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Izydorek, Marek; Marzantowicz, Wacław. The Borsuk-Ulam property for cyclic groups. Topol. Methods Nonlinear Anal. 16 (2000), no. 1, 65--72.

Export citation


  • T. Bartsch, Critical orbits of invariant functionals and symmetry breaking , Manuscripta Math., 66 (1989), 129–152 \ref ––––, On the genus of representation spheres , Comment. Math. Helv., 65 (1990), 85–95 \ref ––––, Borsuk-Ulam theorems for $p$-groups and counterexamples for non $p$-groups , Heidelberg (1989). preprint ;, Topological Methods for Variational Problems with Symmetries , Lecture Notes in Math., 1560 , Springer-Verlag, Berlin (1993) \ref
  • G. Bredon, Introduction to Compact Transformation Groups, Academic Press (1972) \ref
  • K. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Springer-Verlag, 87 (1982) \ref
  • M. Clapp and D. Puppe, Invariants of the Lusternik–Schnirelman type and the topology of critical sets , Trans. Amer. Math. Soc., 298 (1986), 603–620 \ref ––––, Critical point theory with symmetries , J. Reine Angew. Math., 418 (1991), 1–29 \ref
  • E. Fadell, The equivariant Lusternik–Schnirelman method for invariant functionals and relative cohomological index theory , Topological Methods in Nonlinear Analysis, Sem. Math. Sup., 95 , Presses Univ., Montreal (1985), 41–76 \ref
  • P. Hilton and S. Stammbach, A course in Homological Algebra, Springer-Verlag, Graduate Texts in Mathematics, 4 (1971) \ref
  • W. Y. Hsiang, Cohomological Theory of Topological Transformation Groups, Springer-Verlag (1975) \ref
  • M. Izydorek and W. Marzantowicz, Equivariant maps between cohomology spheres , Topol. Methods Nonlinear Anal., 5 (1995), 279-290 \ref
  • W. Marzantowicz, A $G$-Lusternik–Schnirelman category of space with an action of a compact Lie group , Topology, 28 (1989), 403–412 \ref ––––, Borsuk–Ulam theorem for any compact Lie group , J. London Math. Soc. (2), 49 (1994), 195–208 \ref
  • E. Spanier, Algebraic Topology, McGraw–Hill (1966) \ref
  • R. Switzer, Algebraic Topology –- Homotopy and Homology, Springer-Verlag, Die Grundlehren der math. Wiss., Band 212 (1975) \ref
  • T. tom Dieck, Transformation Groups, de Gruyter Studies in Mathematics, 8 (1987) \ref
  • E. Weiss, Cohomology of Groups, Academic Press (1969)