2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166166In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.Differential GeometryAnalysis of PDEs58J35; 58E05text