# Equivariant maps between representation spheres

## TL;DR

This paper establishes conditions for the existence of equivariant maps between representation spheres of a compact Lie group, linking the problem to subgroup fixed point dimensions and Euler class divisibility.

## Contribution

It provides a new criterion based on fixed point dimensions for the existence of equivariant maps between representation spheres, including a reinterpretation via Euler classes for tori.

## Key findings

- Existence of equivariant maps depends on fixed point dimension inequalities.
- Characterization involves divisibility of Euler classes in the torus case.
- Results unify and extend previous understanding of equivariant sphere maps.

## 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 $\dim V^H \leq \dim W^H$ 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.