On the problem of equivariant homotopic classification
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. Laitinen, J. Thornehave.
scientific article; zbMATH DE number 937051