Self-equivalences of dihedral spheres.
One might be tempted to regard papers which only construct examples as of little importance. In the present case, this would be wrong. This paper is a very interesting ``example'' in equivariant self-equivalences. If \(G\) is a finite group, acting on a sphere by orthogonal transformations we have a stabilization map \(I:{\mathcal E}_G(X)\to\{X,x\}_G\). This goes from the \(G\)-homotopy classes of self-equivalences to the (Burnside) ring of stable \(G\)-homotopy classes of self-maps. This displays \({\mathcal E}_G(X)\) as an extension of the group of units of \(\{X,x\}_G\) by the kernel of \(I\). Working with dihedral groups acting on spheres, the author displays an infinite family of \(G\)'s acting on \(X_k\)'s where \(\text{ker\,} I\) is a non-abelian torsion free solvable group, and \(\text{Im\,} I\) is an abelian 2-group of order \(2^k-1\). The torsion free rank of \(\text{ker}\, I\) is calculated and the derived length is estimated. The paper is well-written although I would prefer ``may not be onto'' to ``can not be onto'' in the introduction.
scientific article; zbMATH DE number 1943356