Let \(X\) be a topological space and \(\varepsilon(X)\) denote the group of homotopy classes of self-homotopy equivalences of \(X\). In this paper, the authors compute the nilpotency class of the group \(\varepsilon_\sharp (X)\) consisting of homotopy classes which induce the identity on homotopy groups. They describe the relations between \(\varepsilon_\sharp (X)\) and \(\varepsilon_\sharp (X_0)\), \(X_0\) being the rationalization of \(X\), and they show that very often nil \(\varepsilon_\sharp (X_0) \leq \text{cat }X-1\).
For the entire collection see [Zbl 0832.00046].