Self-equivalences of dihedral spheres. (Q1396460)

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.