2019 Fiscal Year Final Research Report
Research on shadows and fibration structures of 4-manifolds
| Project/Area Number |
19K21019
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| Project/Area Number (Other) |
18H05827 (2018)
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Multi-year Fund (2019)
Single-year Grants (2018) |
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| Review Section |
0201:Algebra, geometry, analysis, applied mathematics,and related fields
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| Research Institution | Chuo University (2019)
Tohoku University (2018) |
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Principal Investigator |
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| Project Period (FY) |
2018-08-24 – 2020-03-31
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| Keywords | ４次元多様体 / シャドウ / レフシェッツファイバー空間 / 微分構造 / 接触構造 / 結び目理論 / 特異点論 / ３次元多様体 |
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| Outline of Final Research Achievements |
We studied differential structures and fibration structures of 4-manifolds by using shadows. The main achievements are as follows. (1) We provided the method of construction of a shadow for a divide and shadow description of Lefschetz fibrations. (2) In terms of shadows, we provided a sufficient condition for an acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. (3) We define shadows for 2-knots and provided some examples. (4) We studied a correspondence between flow-spines and contact structures on 3-manifolds and provided some examples.
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| Free Research Field |
低次元トポロジー
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| Academic Significance and Societal Importance of the Research Achievements |
シャドウは４次元多様体に関する様々な観点を与える；ハンドル分解，``多面体上の''円板束，曲面の埋め込み・はめ込み．今回の研究は，これらの概念を４次元多様体の中で相互に理解しつつ，４次元トポロジーで重要視される微分構造やファイバー構造の研究に対して新たな研究手法を与えるというものである．また，シャドウは多面体という組み合わせ的性質を持つ応用性の高い対象であり，具体例としても扱いやすい側面がある．シャドウの適用例・応用例を提示したことで，今後の低次元トポロジーにおける研究のひとつの方針を与えたと言える．
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