Denote by \(\varepsilon(X)\) the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex \(X\). The authors give results concerning the solvability of certain subgroups of \(\varepsilon(X)\). They prove that the solvability of the subgroup \(\varepsilon_\#(X)\), consisting of homotopy classes of self-homotopy equivalences that induce the identity on the homotopy groups through the dimension of \(X\) is less than the restricted spherical cone-length of \(X\), \(\text{rscl}(X)\). Here \(\text{rscl}(X)\) is the smallest positive integer \(n\) such that there is a sequence of cofibrations \(L_i\to X_i\to X_{i+1}\) for \(0\leq i< n\) such that each \(L_i\) is a finite wedge of spheres, \(\dim L_i< \dim X\), \(X_0\simeq X\) and \(X_n\simeq X\). Then the authors dualize the result and obtain an upper bound for the subgroup of homotopy classes of self-homotopy equivalences that fix cohomology.
The paper contains also interesting comments on solvability, nilpotency and relations with the Gottlieb groups.
For the entire collection see [Zbl 0960.00050].