On the rational homotopical nilpotency index of principal bundles

TL;DR

This paper investigates the rational homotopical nilpotency index of automorphism spaces of principal G-bundles, establishing bounds related to the rational homotopy groups of G and the triviality of the bundle.

Contribution

It provides new inequalities relating the homotopical nilpotency index of automorphism groups to the rational homotopy groups of the structure group G.

Findings

Established bounds for the homotopical nilpotency index in terms of G's rational homotopy groups.

Proved that the index equals the number of non-trivial rational homotopy groups when the bundle is trivial.

Applied results to finite base spaces, showing exact equality in specific cases.

Abstract

Let $\rm{Aut}(p)$ 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 $$n-{\rm N}(p)\leq {\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_0)\leq n$$ for any space $X$, where $n$ is the number of non-trivial rational homotopy groups of $G$ and ${\rm N}(p)$ is defined in Section 2. In particular, ${\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_{0})=n$ if $p$ is a fibre-homotopy trivial bundle and X is finite.