Dimension theory and coarse geometry from the view point of embedding problems

2023 Fiscal Year Final Research Report

• Project/Area Number: 16K05160
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Waseda University
• Principal Investigator: Koyama Akira 早稲田大学, 理工学術院, 名誉教授 (40116158)
• Project Period (FY): 2016-04-01 – 2024-03-31
• Keywords: 次元論 / コホモロジー次元論 / coarse幾何学 / 漸近次元

Outline of Final Research Achievements

We investigated topological properties of ideal boundaries of geodesic spaces from the following two points of view. The first case is that the local homology groups of the boundary is trivial. The second case is that th e boundary has pathological complexity.
In the first case we suppose that the boundary should have similar properties to ANR. Related to this topics I published the joined paper with professor V. Valov (Nipissing University, Canada) from Topology and its Applications.
In the second case we suppose that cohomological dimension theory should be a key tool. Then we analized the classical example by Boltyanski and Kodama and succeeded to simplify the construction. We presented its process at he meeting at RIMS Institute for Mathematical Sciences Kyoto University, 2023, 06, 05 -07.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

M. Gromowが幾何学的群論の新しい攻略法として提唱したcoarse幾何学では測地線空間の理想境界の位相的性質を調べることが重要になる。特に幾何的に解明可能が難しい場合の研究はそれほど進んでいない。本研究はそのような場合に有効と考えられるコホモロジー次元論からの研究とホモロジー論的なANR理論の開発を提唱した. 現段階ではブレイクスルーを与えるまでに至っていないが, 取りかかりとしては着実な進歩を与えた.