Simplifying stable mappings into the plane from a global viewpoint

Kobayashi, Mahito; Saeki, Osamu

doi:10.1090/S0002-9947-96-01576-0

Simplifying stable mappings into the plane from a global viewpoint
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by Mahito Kobayashi and Osamu Saeki PDF

Trans. Amer. Math. Soc. 348 (1996), 2607-2636 Request permission

Abstract:

Let $f : M \to \mathbf {R}^{2}$ be a $C^{\infty }$ 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.

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