Equivariant maps between representation spheres

Abstract

Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that VG=WG={0}, then a G-equivariant map S(V)→S(W) exists provided that dimVH≤dimWH for any closed subgroup H⊆G. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when G is a torus.