On the problem of equivariant homotopic classification
type: Item
instance of: scholarly article
title: On the problem of equivariant homotopic classification
zbMATH Open document ID: 0864.55009
DOI: 10.1007/BF01194175
author: Zalman Balanov
author: A. Kushkuley
published in: Archiv der Mathematik
publication date: 1997-06-23T00:00:00Z
review text: This paper has four sections. First the authors recall some facts on a classical theorem of H. Hopf and on obstruction theory, then they formulate their version of the equivariant Hopf theorem and prove it in Section 4. Making the following assumptions (i) \(G\) is a compact Lie group, \(X\) is a \(G\)-complex of a finite orbit type and \(Y\) is a \(G\)-space (ii) For every \(H\in\text{Iso}(X)\) the space \(X^H\) is a finite-dimensional \(W(H)\)-complex of topological dimension \(n(H)>0\); the space \(Y^H\) is locally and globally \(k\)-connected for each \(k=0,1,\dots, n(H)-\dim W(H)\), if \(|W(H)|=\infty\) and locally and globally \(k\)-connected for each \(k=0,1, \dots,n(H)-1\) otherwise (iii) Let \(\Phi(G,X,Y)= \{H \in\text{Iso} (X):W(H)\) finite, \(\pi_{n(H)} (Y^H)\neq 0\}\) and for \(H\in\Phi(G,X,Y)\) they assume that \(\pi_{n (H)} (Y^H)=Z\) and \(H^{n(H)} (Y^H)=Z\). Let denote the generators \(O_H\in H^{n(H)} (Y^H)\) the authors prove: (a) the set \([X;Y]_G\) of the equivariant homotopy classes is not empty; (b) Let \(H\in\Phi (G,X,Y)\), \(\dim(X^{>H}) <n(H)\) and \(f_1^H\), \(f_2^H: X^H\to Y^H\) are \(W(H)\)-equivariant maps such that \(f_1^H|X^{>H}\), \(f_2^H|X^{>H}\) are \(W(H)\)-equivariantly homotopic. Then \((f_1^H)^*(O_H) \equiv (f_2^H)^* (O_H) \pmod {w_H \cdot H^{n (H)} (X^H)}\). (c) Let \(f:X^{>H}\to Y^H\) be a \(W(H)\)-equivariant map, let \(H\) be as in (b) and let \(\beta\in H^{n(H)} (X^H)\). Then there exists a pair of \(W(H)\)-equivariant extensions \(f_1^H\), \(f_2^H:X^H\to Y^H\) of the map \(f\) such that \((f_1^H)^*(O_H) \equiv (f_2^H)^* (O_H)+ w_H \cdot\beta\). (d) The statement ``\(f_1,f_2:X\to Y\) are \(G\)-equivariant homotopic iff \((f_1|_{X^H})^* (O_H)\equiv (f_2|_{X^H})^*(O_H)\) for every \(H\in\Phi (G,X,Y)\)'' is equivalent to the following condition: \(\text{Ker} j^*|w_H\cdot H^{n(H)} (X^H,X^{>H})=0\) for every \(H\in\Phi (G,X,Y)\). So they generalize results on equivariant homotopy classification obtained by T. tom Dieck, E. Laitine…
reviewed by: Corina Mohorianu
Mathematics Subject Classification ID: 55P91
zbMATH DE Number: 937051
zbMATH Keywords: equivariant maps
zbMATH Keywords: equivariant Hopf theorem
zbMATH Keywords: equivariant homotopy classes
MaRDI profile type: Publication
cites work: On the winding number and equivariant homotopy classes of maps of manifolds with some finite group actions
cites work: Q5658847
cites work: Transformation groups and representation theory
cites work: Transformation groups
cites work: Q4112654
full work available at URL: https://doi.org/10.1007/bf01194175
OpenAlex ID: W1985621667
description: scientific article; zbMATH DE number 937051