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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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Full Text: Euclid arXiv