2023 Fiscal Year Final Research Report
Research on differential and geometric structures of 3- and 4-manifolds with polyhedra
| Project/Area Number |
20K03588
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Keio University (2023)
Hiroshima University (2020-2022) |
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Principal Investigator |
KODA Yuya 慶應義塾大学, 経済学部(日吉), 教授 (20525167)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 3次元多様体 / 接触構造 / 多面体 / スパイン / フロー |
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| Outline of Final Research Achievements |
We carried out the following research using 2-dimensional polyhedra called spines and shadows, which combinatorially describe differentiable 3- and 4-manifolds. Focusing on the correspondence between the Reeb flows of contact structures and positive flow spines, we proved the existence of contact structures for positive flow spines. Further, we showed the surjectivity of the map from the set of positive flow spines to the set of contact structures given by this correspondence. We also gave explicit representations of the fundamental groups of the complements of shadows. On a related topics, we have studied the Goeritz groups of Heegaard splittings, and the homotopy classification of global defects.
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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 |
3次元および4次元多様体論においては，可微分構造を介して定義される諸概念が，組合せ的対応物により解釈され，それにより新たな視覚的・構成的手法が開発されることで，研究が大いに進展してきた．本研究では，ここにおける「組合せ的対応物」としてスパインとシャドウという2次元多面体に着目して研究を行い，特に，「接触構造」と「正フロースパイン」の対応，「特異点」と「シャドウ」の対応の存在を明示的に記述した．前者の対応により，接触構造の Reeb フローの力学系をフロースパイン上の離散力学系として捉えることが原理的に可能になり，後者の対応によりディバイド絡み目の双曲体積の評価が可能になった．
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