On the number of singularities of a generic surface with boundary in a 3-manifold

We consider a C^∞ generic map f: M → N of a compact surface M with boundary into a 3-manifold N with boundary which is neat (i.e., f⁻¹(∂ N) = ∂ M). The isolated singularities of the image f(M) are triple points, cross caps and boundary double points. Under certain homological conditions, we give some formulae relating the numbers of these singularities. We also obtain some geometrical applications of these results.