Simplifying stable mappings into the plane from a global viewpoint

Mahito Kobayashi, Osamu Saeki

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)2607-2636
Number of pages30
JournalTransactions of the American Mathematical Society
Volume348
Issue number7
Publication statusPublished - 1996
Externally publishedYes

Fingerprint

Stable Map
Fibers
Surgery
Fiber
Quotient Space
Local Structure
Diffeomorphism
Connected Components
Simplification
Completion
Critical value
n-dimensional
Torus
Simplify
Refinement
Configuration
Arbitrary
Theorem

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 348, No. 7, 1996, p. 2607-2636.

Research output: Contribution to journalArticle

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