The rational homotopical nilpotency of principal \(U_n(\mathbb C)\)-bundles (Q390422)

Let \(p : E \to B\) be a principal \(U_n(\mathbb{C})\)-bundle. Let \(\mathrm{Aut}(p)\) denote the monoid of fibre homotopy equivalences of \(p\). The homotopical nilpotency of a group-like space was introduced in [\textit{I. Berstein} and \textit{T. Ganea}, Ill. J. Math. 5, 99--130 (1961; Zbl 0096.17602)] and corresponds to the length of the longest non-nullhomotopic commutator map. The rational homotopical nilpotency is obtained by rationalizing the group-like space. The authors give a lower bound for the rational homotopical nilpotency of \(\mathrm{Aut}(p)\) expressed in terms of the number of vanishing characteristic classes of the bundle \(p\). When \(B = \mathbb{C}P^m\) for \(m \geq n-2\), they prove the rational homotopical nilpotency is \(n-1\) if the bundle is not rationally trivial and \(n\) otherwise. The proofs use the model for \(\mathrm{Aut}(p)\) constructed in [\textit{Y. Félix} et al., Homology Homotopy Appl. 12, No. 2, 371--400 (2010; Zbl 1214.55011)].