2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62639We show that if $K$ is a nilpotent finite complex, then $ΩK$ can be built from spheres using fibrations and homotopy (inverse) limits. This is applied to show that if ${\mathrm {map}}_*(X,S^n)$ is weakly contractible for all $n$, then ${\mathrm {map}}_*(X,K)$ is weakly contractible for any nilpotent finite complex $K$.9 pagesAlgebraic Topology55Q05text