Equivariant degree for abelian actions. I: Equivariant homotopy groups
The paper is devoted to the study of \(\Gamma\)-equivariant homotopy groups of spheres in the case of an abelian group \(\Gamma\). In part II the authors compute the degree and apply it to different examples: bifurcation, existence of solutions with different symmetry type and symmetry breaking. In order to study the above mentioned group, they show that for each isotropy subgroup \(H\), there is a fundamental cell in \(B^ H\), of dimension \(\dim V^ H - \dim(\Gamma/H)\), which generates \(B^ H\) and where the action is free. If this dimension is less than \(\dim W^ H\) then they show (in Section 3) that any \(\Gamma\)-map has an equivariant extension to \(B^ H\), while if the dimensions are equal, then there is an obstruction, an integer which is the degree of an extension of the map to the boundary of the cell. It is proved in Section 4 that if \(\Gamma\) preserves orientations this obstruction is unique and well defined. Conditions under which the obstruction is independent of the previous extensions are given. In section 5 they decompose the homotopy group into subgroups corresponding to Weyl groups of fixed dimension \(k\). If for each such isotropy subgroup \(H\), the dimension of its fundamental cell is less than or equal to \(\dim W^ H\), then the subgroup is a product of \(\mathbb{Z}\)'s, one for each \(H\) where one has equality. In section 6 the relationship between the different possible equivariant degree is studied. They give an explicit relation, if \(\dim V^ H = \dim W^ H\), between these equivariant degrees and the usual degrees of restrictions of the map to isotropy subspaces. Explicit generators for different pieces of equivariant homotopy groups are given (section 7). Section 8 is devoted to a complete characterization of the homotopy group when \(k = 1\). In this case the torsion part of this group is a finite group generated by explicit maps and relations. The final section gives an equivariant suspension theorem which is necessary for the extension to infinite dimensions. The authors also prove that any element in the homotopy group is the \(\Gamma\)-degree of a map on \(\Omega \subset V\), in the cases \(k = 0\) or 1 or \(\dim \Gamma/H = k\).
scientific article; zbMATH DE number 591500