### Abstract

Let f : M → R^{2} 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.

Original language | English |
---|---|

Pages (from-to) | 2607-2636 |

Number of pages | 30 |

Journal | Transactions of the American Mathematical Society |

Volume | 348 |

Issue number | 7 |

Publication status | Published - 1996 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*348*(7), 2607-2636.

**Simplifying stable mappings into the plane from a global viewpoint.** / Kobayashi, Mahito; Saeki, Osamu.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 348, no. 7, pp. 2607-2636.

}

TY - JOUR

T1 - Simplifying stable mappings into the plane from a global viewpoint

AU - Kobayashi, Mahito

AU - Saeki, Osamu

PY - 1996

Y1 - 1996

N2 - 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 Wf 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.

AB - 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 Wf 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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M3 - Article

VL - 348

SP - 2607

EP - 2636

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -