  Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal
$G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V)
\to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup
$H\subseteq G$. This result is complemented by a reinterpretation in terms of
divisibility of certain Euler classes when $G$ is a torus.
