On the rational homotopical nilpotency index of principal bundles

Abstract

Let $\mathrm{Aut}(\mathrm{p})$ denote the space of all self-fibre homotopy equivalences of a principal $G$-bundle $p:E\to X$ of simply connected CW complexes with $E$ finite. When $G$ is a compact connected topological group, we show that there exists an inequality $$n-\mathrm{N}(p)\le {\mathrm{Hnil}}_{\mathbb{Q}}(\mathrm{Aut}(p{)}_0)\le n$$ for any space $X$, where $n$ is the number of non-trivial rational homotopy groups of $G$ and $\mathrm{N}(p)$ is defined in Section 2. In particular, ${\mathrm{Hnil}}_{\mathbb{Q}}(\mathrm{Aut}(p{)}_0)=n$ if $p$ is a fibre-homotopy trivial bundle and X is finite.