Simplifying stable mappings into the plane from a global viewpoint

Authors:
Mahito Kobayashi and Osamu Saeki

Journal:
Trans. Amer. Math. Soc. **348** (1996), 2607-2636

MSC (1991):
Primary 57R45; Secondary 57R35, 57M99

DOI:
https://doi.org/10.1090/S0002-9947-96-01576-0

MathSciNet review:
1344209

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Abstract: Let be a stable map of an -dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold . For this purpose, we also study the quotient space of , which is the space of the connected components of the fibers of , and we completely determine its local structure for arbitrary dimension of the source manifold . 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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Additional Information

**Mahito Kobayashi**

Affiliation:
Department of Mathematics, Akita University, Akita 010, Japan

Email:
mahito@math.akita-u.ac.jp

**Osamu Saeki**

Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739, Japan

Email:
saeki@top2.math.sci.hiroshima-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-96-01576-0

Received by editor(s):
October 24, 1994

Additional Notes:
The second author has been partially supported by CNPq, Brazil, and by Grant-in-Aid for Encouragement of Young Scientists (No. 07740063), Ministry of Education, Science and Culture, Japan

Article copyright:
© Copyright 1996
American Mathematical Society