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 $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V)\to S(W)$ exists provided that $\dim V^H\le \dim W^H$ 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.

The previous Subsection 4.1 and Section 5 are removed, as they relied on a false result from another paper: it is not true that the localization of the integral cohomology ring of the classifying space of $G=({\mathbb{S}}^1{)}^k\times ({\mathbb{Z}}_p{)}^l$ with respect to the set of Euler classes of complex $G$-representations without a trivial direct summand is non-zero. 10 pages, no figures