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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 $f : M \to \text {\bf R}^{2}$ 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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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

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