The Borsuk-Ulam property for cyclic groups

An orthogonal representation $V$ of a group $G$ is saidto have the Borsuk-Ulam property ifthe existence of an equivariant map$f:S(W) \rightarrow S(V)$ from a sphere of representation $W$ intoa sphere of representation $V$implies that $\dim W \leq \dim V$. It is known thata sufficient condition for $V$ to have the Borsuk-Ulam propertyis 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 $G$-categoryof a $G$-space $X$ is also derived.