Equivariant maps between representation spheres
| Autor: | Wacław Marzantowicz, Zbigniew Błaszczyk, Mahender Singh |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
General Mathematics equivariant map Borsuk–Ulam theorem 01 natural sciences symbols.namesake representation sphere FOS: Mathematics Algebraic Topology (math.AT) Mathematics - Algebraic Topology 0101 mathematics 55S37 55S35 Mathematics Euler class 010102 general mathematics Representation (systemics) Lie group Torus Divisibility rule Borsuk--Ulam theorem 55N91 010101 applied mathematics 55M20 Euler's formula symbols Equivariant map 55S37 (primary) 55M20 55S35 55N91 (secondary) |
| Zdroj: | Bull. Belg. Math. Soc. Simon Stevin 24, no. 4 (2017), 621-630 |
| DOI: | 10.48550/arxiv.1704.01656 |
| Popis: | 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. Comment: 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 |
| Databáze: | OpenAIRE |
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