Self homotopy equivalences of equivariant spheres (Q2734860)

Let \(G\) be a finite group, \(V\) an orthogonal real representation of \(G\), \(S(V)\) the unit sphere in \(V\) and \(\varepsilon_G(S(V))\) the group of units in the monoid \([S(V), S(V)]_G\) of free equivariant homotopy classes of self-maps. The purpose of the paper is the analysis of the kernel and the image of the equivariant degree homomorphism NEWLINE\[NEWLINEd_G: \varepsilon_G(S(V))\to \prod_{\text{Iso}(S(v))}\mathbb{Z}.NEWLINE\]NEWLINE The author proves that the kernel is a torsion-free, finitely generated and solvable subgroup, and the image an elementary Abelian 2-group. He gives very precise information on the rank of kernel and dimension of image. The paper contains also explicit computation on some examples.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00050].