Nilpotency of groups of fibre homotopy equivalences in the rational category (Q1867166)

In the paper under review, the author extends previous results [\textit{Y. Félix} and \textit{A. Murillo}, Bull. Lond. Math. Soc. 29, 486-488 (1997; Zbl 0881.55009)] on estimation of the nilpotency index of some groups of homotopy self-equivalences. He gives an upper bound for the group of fiber-homotopy fiber-self equivalences of a fibration inducing the identity on the homotopy groups of the fibre. Dualizing, the author considers the group \(\varepsilon^m_*(X)\) formed by the self-homotopy equivalences of \(X\) inducing the identity in homology up to degree \(m\). He proves that nil \(\varepsilon^m_* (X) \leq\text{cocat} X-1\) if \(\dim X\leq m\) and gives an estimate for sol \(\varepsilon^m_* (X)\).