M. Kobayashi et O. Saeki, SIMPLIFYING STABLE MAPPINGS INTO THE PLANE FROM A GLOBAL VIEWPOINT, Transactions of the American Mathematical Society, 348(7), 1996, pp. 2607-2636
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.