On the self-intersection set and the image of a generic map

Let f:M→N be a continuous map of a closed m-dimensional manifold into an n-dimensional manifold with k=n-m>0. We define a primary obstruction to the existence of a homotopy between f and a smooth embedding which is related to the self-intersection set of a generic map homotopic to f. When f is a smooth generic map in the sense of [9], we show that f is a smooth embedding if and only if the primary obstruction vanishes and the (m-k+1)th Betti numbers of M and the image f(M) coincide, generalizing the authors' previous result [4] for immersions with normal crossings. As a corollary we obtain a converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results in [3, 1, 2, 20] for immersions with normal crossings. Using generic maps, we show the vanishing of the primary obstruction for injective maps. Furthermore, for non-generic smooth maps, we find a homology class in the closure of the self-intersection set which corresponds to the primary obstruction.