On the Betti number of the image of a generic map

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Let f : M ! N be a dierentiable map of a closed m-dimensional manifold into an (m + k)-dimensional manifold with k> 0. We show, assuming 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 homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings (BS1). As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of (BR, BMS1, BMS2, Sae1) for immersions with normal crossings.