Aspherical completions and rationally inert elements

Abstract

Let $X$ be a connected space. An element $[f]\in \pi_n(X)$ is called rationally inert if $\pi_*(X)\otimes \mathbb Q \to \pi_*(X\cup_fD^{n+1})\otimes \mathbb Q$ is surjective. We extend the results obtained in the simply connected case, and prove in particular that if $X\cup_fD^{n+1}$ is a Poincar\'e duality complex and the algebra $H(X)$ requires at least two generators then $[f]\in \pi_n(X)$ is rationally inert. On the other hand, if $X$ is rationally a wedge of at least two spheres and $f$ is rationally non trivial, then $f$ is rationally inert. Finally if $f$ is rationally inert then the rational homotopy of the homotopy fibre of the injection $X \to X\cup_fD^{n+1}$ is the completion of a free Lie algebra.