Equivariant maps between representation spheres

Zbigniew Błaszczyk; Wacław Marzantowicz; Mahender Singh

Equivariant maps between representation spheres

Algebraic Topology 2018-01-09 v2

Authors: Zbigniew Błaszczyk , Wacław Marzantowicz , Mahender Singh

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.

Keywords

equivariant cohomology group representations mapping theory

Cite

@article{arxiv.1704.01656,
  title  = {Equivariant maps between representation spheres},
  author = {Zbigniew Błaszczyk and Wacław Marzantowicz and Mahender Singh},
  journal= {arXiv preprint arXiv:1704.01656},
  year   = {2018}
}

Comments

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

Equivariant maps between representation spheres

Abstract

Keywords

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