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 ${\pi }_{\ast }(X)\otimes \mathbb{Q}\to {\pi }_{\ast }(X{\cup }_f{D}^{n+1})\otimes \mathbb{Q}$ is surjective. We extend the results obtained in the simply connected case, and prove in particular that if $X{\cup }_f{D}^{n+1}$ is a Poincaré 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 $X\to X{\cup }_f{D}^{n+1}$ is the completion of a free Lie algebra.

References in corpus (2)

• Algebraic invariants for right-angled Artin groups
• Sullivan completions