2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62345We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.Differential GeometryMetric Geometry53C20text