Finitely presented subgroups of the self-homotopy equivalences group (Q1922568)

Let \(X\) be a pointed space and \({\mathcal E} (X)\) the group of homotopy classes of self-equivalences (pointed) of \(X\). It is now known that under very broad conditions, \({\mathcal E} (X)\) is finitely presented [see, for example, \textit{E. Dror}, \textit{W. G. Dwyer} and \textit{D. M. Kan}, Comment. Math. Helv. 56, 599-614 (1981; Zbl 0504.55004)]. The present author studies this problem for certain important subgroups of \({\mathcal E} (X)\). For example, the author shows that when \(X\) is a finite nilpotent space, the centralizer of a finite subset or a finitely generated subgroup of \({\mathcal E} (X)\) is finitely presented. It is also shown that under these conditions, any nilpotent subgroup of \({\mathcal E} (X)\) is finitely presented. The methods include minimal models and algebraic groups. Details are too complex to give here.