Resolution of corank 1 singularities in the image of a stable smooth map to a space of higher dimension

We consider stable smooth maps from closed smooth manifolds to smooth manifolds of higher dimension. For maps with corank 1 singularities, we find universal linear relations between the Euler characteristics of the manifolds of singularities in their images. The calculations are based on resolving the singularities by a construction that generalizes the iteration principle from algebraic geometry.