2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226258We find conditions on topological spaces X, Y and nonempty subset B of Y which guarantee that for each continuous map f from X to Y there exists a map g homotopic to f such that Nielsen preimage classes of g^{-1}(B) are all topologically essential.This is the version published by Geometry & Topology Monographs on 29 April 2008Algebraic Topology54H99, 55M99, 55S35text