Simple stable maps of 3-manifolds into surfaces

Saeki, Osamu

doi:10.1016/0040-9383(95)00034-8

Cited by 32 publications

(100 citation statements)

References 16 publications

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“…Thus we may assume that each point of W has a conic neighborhood as in Figure 1 (a) or (b). Since it has a structure of arm and stem, W is a branched surface in the terminology of [Sa2]. Then we can represent W by its associated graph G W (for details, see [Sa2,§3]).…”

Section: Proposition 52 Let W Be a Compact Pseudo Quotient Space Sumentioning

confidence: 99%

“…Furthermore, it is oriented, once an orientation of R 2 is fixed. Furthermore, a branched surface as defined in [Sa2] is a pseudo quotient space in the above sense. Definition 3.2.…”

Section: Pseudo Quotient Maps and Configuration Trivialitymentioning

confidence: 99%

“…For example, a fold map as defined in [Sa2] is a pseudo quotient map in the sense of Definition 3.2.…”

Section: Pseudo Quotient Maps and Configuration Trivialitymentioning

confidence: 99%

“…In order to define an R-operation in a general context, we will need to define a pseudo quotient space W and a pseudo quotient map q : M → W , which are generalizations of the quotient space W f and the quotient map q f : M → W f of a stable map f : M → R 2 respectively (see, for example, [BdR,L2,KLP,Sa2]). …”

Section: Introductionmentioning

confidence: 99%

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Simplifying stable mappings into the plane from a global viewpoint

Kobayashi

Saeki

1996

Trans. Amer. Math. Soc.

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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.

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Section: Proposition 52 Let W Be a Compact Pseudo Quotient Space Sumentioning

confidence: 99%

“…Furthermore, it is oriented, once an orientation of R 2 is fixed. Furthermore, a branched surface as defined in [Sa2] is a pseudo quotient space in the above sense. Definition 3.2.…”

Section: Pseudo Quotient Maps and Configuration Trivialitymentioning

confidence: 99%

“…For example, a fold map as defined in [Sa2] is a pseudo quotient map in the sense of Definition 3.2.…”

Section: Pseudo Quotient Maps and Configuration Trivialitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Simplifying stable mappings into the plane from a global viewpoint

Kobayashi

Saeki

1996

Trans. Amer. Math. Soc.

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“…According to [Saeki 1996] a closed orientable 3-manifold M has a stable map into ‫ޒ‬ 2 without singular points of types (C), (D) and (E) if and only if M is a graph manifold. By [Levine 1965] a 3-manifold always has a stable map into ‫ޒ‬ 2 without singular points of type (C).…”

Section: Introductionmentioning

confidence: 99%

Maps on 3-manifolds given by surgery

Kalmar¹,

Stipsicz²

2012

Pacific J. Math.

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Stable maps and branched shadows of 3-manifolds

Ishikawa

Koda

2016

Math. Ann.

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Abstract. Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the real plane, and prove that this number equals the minimal number of vertices of its branched shadows. In consequence, we give a complete characterization of hyperbolic links in the 3-sphere whose exteriors have stable map complexity 1 in terms of Dehn surgeries, and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes.

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Simple stable maps of 3-manifolds into surfaces

Cited by 32 publications

References 16 publications

Simplifying stable mappings into the plane from a global viewpoint

Simplifying stable mappings into the plane from a global viewpoint

Maps on 3-manifolds given by surgery

Stable maps and branched shadows of 3-manifolds

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