2022 Fiscal Year Final Research Report
Study of group actions on manifolds by psedo-inverse limit systems of equivariant framed maps
| Project/Area Number |
18K03278
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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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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Okayama University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
早坂 太 岡山大学, 環境生命科学研究科, 准教授 (20409460)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 多様体上の群作用 / 枠付き同変写像 / 同変手術 / 球面上の群作用 |
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| Outline of Final Research Achievements |
Let G be a finite group, A a set of subgroups of G, and B the set of subgroups not belonging to A. We say that a G-action on Z is B-free if the L-fixed-point set of Z coincides with the G-fixed-point set of Z. We consider pseudo-inverse-limit systems F_M : W_M -> Ix Y, where M runs over A, between a G-map f : X -> Y and id : Y -> Y. Choosing a suitalbe pseudo-inverse-limit system and performing G-surgeries, we would discover new G-actions on the underlying manifold of Y. Studying this problem, we could determine the dimension of spheres S with B-free G-action such that S has exactly one G-fixed point for groups: Alternating Groups A_5, A_6 (degree 5, 6), Symmetric Group S_5 (degree 5), Double Covering Groups SL(2, 5) of A_5 and TL(2, 5) of S_5, and etc.
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| Free Research Field |
微分位相幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
有限群 G が多様体 X, Y に作用している状況で，G-写像 f : X -> Y を同変手術により微分同相写像にホモトピックな f' : X' -> Y に変形する問題は難しい問題である．特に，ある部分群 H に対し X のH-不動点集合の次元が 3, 4 となる場合には極めて難しいい問題である．本課題研究では f と恒等写像 id の間の枠付きM-コボルディズム F_M : W_M -> I x Y（M を A 上で動かす）の擬逆極限系をうまく選んでこの困難さを克服する研究を行い，うまい選択方法を得ることができた．ここに学術的意義がある．
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