## Topological Methods in Nonlinear Analysis

### The Borsuk-Ulam property for cyclic groups

#### Abstract

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

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

Dates
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.tmna/1471875422

Mathematical Reviews number (MathSciNet)
MR1805039

Zentralblatt MATH identifier
0998.57059

#### Citation

Izydorek, Marek; Marzantowicz, Wacław. The Borsuk-Ulam property for cyclic groups. Topol. Methods Nonlinear Anal. 16 (2000), no. 1, 65--72. https://projecteuclid.org/euclid.tmna/1471875422

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