| Project/Area Number |
17K05257
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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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| Research Field |
Geometry
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | トポロジー / ３次元多様体 / カンドル / 結び目 / 幾何学 / べき零 / 数学 |
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| Outline of Final Research Achievements |
The purpose of this study is to analyze quandle, which is an algebraic system, and to give its applications to the field of topology. In this study, I investigate quandles from the viewpoints of manifolds and nilpotent methods, and obtained some results. Meanwhile, given a quandle, a (co)-homology theory is defined; from the viwpoints, I calculate quandle cohomology with smoothness and the second cohomology.
In applications to topology, I gave a procedure of computing the fundamental 3-class of a 3-dimensional manifold. If the manifold is a knot complement or we are give a nilpotent setting, I showed that the procedure can be described as a concrete formula. I also gave some results on centrality of meta-nilpotent quotient groups of the free group.
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| Academic Significance and Societal Importance of the Research Achievements |
数学という分野は基礎学問にして、多くの理論発展や実用的技術などにも(間接的ながらも)影響をもたらしてきた。その中、筆者の分野は幾何学、特に3次元トポロジーを扱っている。我々は3次元空間内に生活しているが、3次元的オブジェは複雑な様相と幾何構造を擁し、未解明な点が多い。筆者の専門分野の一つとして紐の結ばり方は、想像以上の複雑さを擁し、数学的に興味深い研究手法が適用可能である。多くの研究手法のうち、カンドルや基本群やカップ積という手法を筆者は扱っている。これらの手法は幾何学の中で歴史あるもので、その発展は学術的意義があるものと考えている。
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