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.