ON THE BETTI NUMBER OF THE IMAGE OF A GENERIC MAP

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.