2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126994We will prove the relative homotopy principle for smooth maps with singularities of a given {\cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {\cal K}-codimensions.Typographic errors are revised and several expressions are modified to better onesGeometric Topology58K30; 57R45text