2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144794The goal of this paper is to ameliorate the sufficients conditions, already established by the first author so that the sum of the numbers of Betti, of 1-connected rational finite CW-complex, is higher than the dimension of his $\mathbb Q$-vectorial space of homotopy, we will present it in two aspects, one algebraic and another geometrical.18 pages; Work exposed by the second author in the Algebraic topology conference of Angers, France, October 2007Algebraic Topology55N34; 55P62; 57T99text