2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/145821We determine the cup-length of some oriented Grassmann manifolds by finding a Groebner basis associated with a certain subring of the cohomology of them. As its applications, we provide not only a lower but also an upper bound for the LS-category of some oriented Grassmann manifolds. We also study the immersion problem of them.Corrected errors: Conjecture 1.1Algebraic Topology55M30text