Simplifying stable mappings into the plane from a global viewpoint

Let f : M → R² be a C^∞ stable map of an n-dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on f which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold M. For this purpose, we also study the quotient space W_f of f, which is the space of the connected components of the fibers of f, and we completely determine its local structure for arbitrary 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 Furuya for orientable manifolds of dimension 4. We also pay special attention 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.