| Project/Area Number |
16340018
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | KYUSHU UNIVERCITY |
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Principal Investigator |
SAEKI Osamu Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (30201510)
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| Co-Investigator(Kenkyū-buntansha) |
SAKUMA Kazuhiro Kinki University, School of Science and Engineering, Associate Professor, 理工学部, 助教授 (80270362)
OHMOTO Toru Hokkaido University, Faculty of Science, Associate Professor, 大学院理学研究院, 助教授 (20264400)
IWASE Norio Kyushu University, Faculty of Mathematics, Associate Professor, 数理学研究院, 助教授 (60213287)
NISHI Haruko Kyushu University, Faculty of Mathematics, Research Associate, 数理学研究院, 助手 (90274430)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥6,600,000 (Direct Cost: ¥6,600,000)
Fiscal Year 2006: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2005: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥2,600,000 (Direct Cost: ¥2,600,000)
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| Keywords | global theory of singularities / homotopy theory / obstruction / singular fiber / classifying space / universal complex / differentiable structure / manifold / 消去 / ホモトピー論 / 写像芽 / 位相不変量 / トポロジー / 特異点論 / 同境 / ホモトピー不変量 / モース関数 / 符号数 / 可微分構造 / トム多項式 |
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| Research Abstract |
The study of differentiable maps between manifolds and their singularities was begun by Whitney, Thom etc. in the middle of the 20th century and has developed a lot since then. In particular, local properties have been studied and some sophisticated theories have been established for such studies. However, global properties of maps that are essentially related to the structures of manifolds have not been studied so much in spite of their importance. In this research project, we aimed at solving various important open problems in the global theory of singularities from the viewpoint of homotopy theory in a larger framework. More precisely, we performed the following studies and obtained some results, which will be explained below :
(a) Higher obstructions,
(b) Classifying space of singular fibers,
(c) Relationship between the differentiable structures of manifolds and singularities of maps.
As to (a), Saeki and Iwase studied differentiable maps whose regular fibers consist of spheres, and obtained some homotopical properties of those manifolds which admit such maps. Furthermore, it has been clarified that these properties are related to higher obstructions to the existence of such maps. Moreover, Saeki and Sakuma studied the existence problem of fold maps and found that certain Postnikov invariants appear as higher obstructions. As to (b), Saeki constructed characteristic classes of surface bundles by using the theory of singular fibers of functions on surfaces. Furthermore, Saeki and Ohmoto succeeded in constructing a classifying space of singular fibers. As to (c), Saeki and Sakuma collected and arranged the known results about the relationship between the singularities of differentiable maps and differentiable structures of manifolds, and Saeki showed that in certain cases the elimination of definite fold singularities is possible independently of the differentiable structures of manifolds.
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