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Equivariant maps between representation spheres. (English) Zbl 1422.55033
Summary: 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.

MSC:
 55S37 Classification of mappings in algebraic topology 55M20 Fixed points and coincidences in algebraic topology 55S35 Obstruction theory in algebraic topology 55N91 Equivariant homology and cohomology in algebraic topology
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