Simplifying stable mappings into the plane from a global viewpoint

Abstract

Let f : M → R 2 f : M \to \mathbf {R}^{2} be a C ∞ C^{\infty } stable map of an n n -dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on f f which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold M M . For this purpose, we also study the quotient space W f W_{f} of f f , which is the space of the connected components of the fibers of f f , and we completely determine its local structure for arbitrary dimension n n of the source manifold M 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.