2022 Fiscal Year Final Research Report
Study on Fiberwise A-infinity Structures
| Project/Area Number |
19K14535
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Kyushu University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | ホモトピー論 / ファイバーワイズホモトピー論 / A無限大空間 / crossed module |
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| Outline of Final Research Achievements |
The most important result of this project is the development of the theory of higher homotopy normality using fiberwise A-infinity structures. Though higher homotopy normality has been studied by several people, there are no theory of ``essentially higher'' homotopy normality. The theory established in this project is a candidate for such theory. It enables us to determine when a given homomorphism has higher homotopy normality by the classical technique in the fiberwise homotopy theory. Indeed, the p-local higher homotopy normality of the inclusions SU(m) -> SU(n) are determined for some m,n and p.
We also obtained some results on the homotopy type of the unitary groups of some uniform Roe algebras in a joint work. Comparing to Roe algebras, uniform Roe algebras tend to have huge K-theory. We could determine the homotopy type of the unitary groups of uniform Roe algebras on Z and Z^2.
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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 |
高次ホモトピー正規性はこれまでにも研究されてきたが，「本質的に高次の」ホモトピー正規性の理論は得られていなかった．得られた理論では古典的なファイバーワイズホモトピー論の技術を用いて，準同型が高次ホモトピー正規性を持つかどうか調べられる点が強みである．実際，包含写像SU(m) -> SU(n)のp-局所的なホモトピー正規性をいくつかの場合に決定した．このように扱いやすさも実証できており，今後の発展が期待できる．
一様Roe代数のユニタリ群は巨大なホモトピー群を持つ（一様Roe代数のK群と一致）ため難解であるが，実際にホモトピー型を調べる手法を与えた．距離を考慮したトポロジーへの応用も期待できる．
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