On the rational homotopical nilpotency index of principal bundles

Abstract

Let Aut(p) denote the space of all self-fibre homotopy equivalences of a principal G-bundle $p: E\rightarrow X$ of simply connected CW complexes with E finite. When G is a compact connected topological group, we show that there exists an inequality

$$\begin{aligned} n-\mathrm{N}(p)\le \mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_0)\le n \end{aligned}$$

for any space X, where n is the number of non-trivial rational homotopy groups of G and $\mathrm{N}(p)$ is defined in Sect. 2. In particular, $\mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_{0})=n$ if p is a fibre homotopy trivial bundle and X is finite.