2018 Fiscal Year Final Research Report
Study on higher homotopy structures of loop spaces
| Project/Area Number |
16K17592
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Kyushu University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | リー群 / ループ空間 / 写像空間 / 位相的複雑さ / ゲージ群 / 高次ホモトピー可換性 |
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| Outline of Final Research Achievements |
In this project, I studie higher homotopy structures of multiplications on loop spaces such as topological groups. Among the results of this project, I obtained the results on higher homotopy commutativity of Lie groups and topological complexity.
For the former result, I and the collaborators obtained the higher homotopy commutativity of Lie groups stronger than the one already known. As an application, we first succeeded to prove the higher homotopy commutativity of gauge groups.
For the latter result, I and the collaborators studied a certain fiberwise loop structure and provided an easy method to compute the homotopy invariant called the topological complexity of the Klein bottle.
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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 |
高次ホモトピー可換性はLie群の通常の意味での可換性に端を発する問題で，長い間研究されてきた問題である．高次ホモトピー可換性がわかると分類空間の高次Whitehead積などの自明性が導かれるなど，種々の不変量の計算に重要な応用を持つ．
また，ファイバーワイズなループ構造についてはロボット動作設計に起源をもつ位相的複雑さなど，興味深い不変量が関連しているが，ファイバーワイズなループ構造自体の取り扱いが未だによくわかっておらず，まず計算が難しく，計算可能なものでも膨大な計算を必要とするものが多い．本研究では新たな計算方法であって，しかも簡明なものを提供することに成功した．
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