2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119916Let X be a finite CW-complex of dimension q. If its fundamental group $π_{1}(X)$ is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups $π_{i}(X)$ is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space $K(π_{1}(X),1)$.27 pages, 0 figuresGeometric TopologyDifferential Geometry57R70, 55P15, 57Q10text