Let f : M --> N be a differentiable map of a closed m-dimensional mani
fold into an (m + k)-dimensional manifold with k > 0. We show, assumin
g that f is generic in a certain sense, that f is an embedding if and
only if the (m - k + 1)-th Betti numbers with respect to the Cech homo
logy of M and f(M) coincide, under a certain condition on the stable n
ormal bundle of f. This generalizes the authors' previous result for i
mmersions with normal crossings [BS1]. As a corollary. we obtain the c
onverse of the Jordan-Brouwer theorem for codimension-l generic maps,
which is a generalization of the results of [BR, BMS1, BMS2, Sael] for
immersions with normal crossings.