On Transversality with Deficiency and a Conjecture of Sard

Let f : M → N be a Cr map between Cr manifolds (r ≥ 1) and K a Cr manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of Cr maps g : K → N which satisfy a certain transversality condition with respect to f in a weak sense. As an application, by considering the case where K is a point, we obtain some new results about the topological properties of f(Rq(f)), where Rq(f) is the set of points of M where the rank of the differential of f is less than or equal to q. In particular, we show a result about the topological dimension of f(Rq(f)), which is closely related to a conjecture of Sard concerning the Hausdorff measure of f(Rq(f)).