  Let 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.
