SIMPLIFYING STABLE MAPPINGS INTO THE PLANE FROM A GLOBAL VIEWPOINT

Let f : M --> R(2) be a C-infinity Stable map of an n-dimensional mani fold into the plane. The main purpose of this paper is to define a glo bal surgery operation on f which simplifies the configuration of the c ritical value set and which does not change the diffeomorphism type of the source manifold M. For this purpose, we also study the quotient s pace W-f of f, which is the space of the connected components of the f ibers of f, and are completely determine its local structure for arbit rary dimension n of the source manifold M. This is a completion of the result of Kushner, Levine and Porto for dimension 3 and that of Furuy a for orientable manifolds of dimension 4. We also pay special attenti on to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2-sphere, which is a refinement of a result of Kobayashi.