2019 Fiscal Year Final Research Report
Studies of higher twisted cohomology groups of automorphism groups of free groups
| Project/Area Number |
16K05155
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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 University of Science |
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Principal Investigator |
Satoh Takao 東京理科大学, 理学部第二部数学科, 准教授 (70533256)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 自由群の自己同型群 / 群のコホモロジー / 非安定コホモロジー / Andreadakis予想 |
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| Outline of Final Research Achievements |
We obtained three main results. The first one is that we found a new GL-irreducible component in the second homology group of the IA-automorphism group of the free group of rank 3. The second one is that we gave the affirmative answer to the Andreadakis problem in the degree three case. The third one is that we obtained non-trivial unstable twisted second cohomology classes of automorphism groups of free groups by using Kawazumi's 1-cocycles.
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| Free Research Field |
代数的位相幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
近年，自由群の自己同型群のコホモロジーの研究に関しては，非安定域に大きな研究の関心が集まっており，階数3の場合のIA自己同型群のホモロジーの新しいGL既約表現や，自由群の自己同型群の非安定2次元コホモロジー類の構成などは，最近の最先端の研究情勢から鑑みても相応のインパクトがあるものと確信している．
また，Andreadakis予想は50年以上も未解決問題であり，一部分とはいえその解決に寄与できたことは，一般の場合の解決に向けた一つの足掛かりを与えたことでもあり，大変有意義であると考えている．
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