2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228594Let M,N and B\subset N be compact smooth manifolds of dimensions n+k,n and \ell, respectively. Given a map f from M to N, we give homological conditions under which g^{-1}(B) has nontrivial cohomology (with local coefficients) for any map g homotopic to f. We also show that a certain cohomology class in H^j(N,N-B) is Poincare dual (with local coefficients) under f^* to the image of a corresponding class in H_{n+k-j}(f^{-1}(B)) when f is transverse to B. This generalizes a similar formula of D Gottlieb in the case of simple coefficients.This is the version published by Algebraic & Geometric Topology on 4 October 2006Geometric TopologyAlgebraic Topology55M20, 55S35text