On relations between gradient and classical equivariant homotopy groups of spheres
type: Item
instance of: scholarly article
title: On relations between gradient and classical equivariant homotopy groups of spheres
zbMATH Open document ID: 1268.55004
author: Kazimierz Gȩba
author: Marek Izydorek
published in: Journal of Fixed Point Theory and Applications
publication date: 2013-05-13T00:00:00Z
review text: Let \(V\) be a finite-dimensional representation of a compact Lie group \(G\) and let \(f : U \to V\) denote a \(G\)-equivariant map on an open \(G\)-invariant subset \(U \subset \mathbb{R}^k\oplus V\) such that \(f^{-1}(0)\) is compact. If \(f\) satisfies the equality \(f(x, v)=\nabla_v\varphi(x, v)\) where \(\varphi : U \to \mathbb{R}\) is a \(G\)-invariant \(C^1\)-function, then it is called a \(G\)-equivariant gradient map. This map \(f\) extends continuously as a map \(\hat{f} : S^{k+V} \to S^V\) between their one-point compactifications. This paper is concerned with the problem of whether or not any two equivariant gradient maps belonging to the same stable equivariant homotopy class are stably homotopic in the category of gradient equivariant maps. The authors examine this problem by enlarging the category of equivariant maps considered there and in fact the authors prove that under a certain condition its assertion holds true for orthogonal equivariant maps. Here by a \(G\)-equivariant orthogonal map is meant a \(G\)-map \(f : U \to V\) satisfying the condition \(f(x, v) \bot T_vGv\). Let \(S_e^{k+V}\) denote a subset of \(S^{k+V}\) on which \(G\) acts freely. Put \(A^{k+V}=S^{k+V}\setminus S_e^{k+V}\) and write \([S^{k+V}, A^{k+V}; S^V, *]_G^\bot\) for the set of relative homotopy classes of orthogonal equivariant maps. Then the principal conclusion of this paper is that if \(q=\dim G \leq \dim V-k-2\), then there is a natural bijection from \([S^{k+V}, A^{k+V}; S^V, *]_G^\bot\) to the set \([S^{q+k+V}, A^{q+k+V}; S^V, *]_G^*\) of relative homotopy classes of equivariant maps.
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55R70
zbMATH DE Number: 6162209
zbMATH Keywords: equivariant gradient maps
zbMATH Keywords: stable homotopy groups
zbMATH Keywords: fiberwise maps
MaRDI profile type: Publication
full work available at URL: s11784-013-0105-8
OpenAlex ID: W2048042111
Wikidata QID: Q59289501
cites work: The topology of torus actions on symplectic manifolds. Transl. from the French by the author
cites work: Otopy classes of equivariant maps
cites work: Vector fields and transfers
cites work: Q5658847
cites work: Q4212657
cites work: A new degree for \(S^ 1\)-invariant gradient mappings and applications
cites work: Classification of homotopy classes of equivariant gradient maps
cites work: Äquivariante Homotopie. I
cites work: A decomposition formula for equivariant stable homotopy classes
cites work: The Burnside ring of a compact Lie group. I
cites work: Transformation groups
cites work: Transformation groups and representation theory
DOI: 10.1007/S11784-013-0105-8
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description: scientific article; zbMATH DE number 6162209