  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.
