| Project/Area Number |
15340025
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyushu University |
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Principal Investigator |
IWASE Norio Kyushu University, Faculty of Mathematics, Associate Professor, 大学院数理学研究院, 助教授 (60213287)
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| Co-Investigator(Kenkyū-buntansha) |
KAMATA Masayoshi Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (60038495)
SAEKI Osamu Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (30201510)
SUMI Toshio Kyushu University, Faculty of Mathematics, Assosiate Professor, 大学院芸術工学院, 助教授 (50258513)
ODA Nobuyuki Fukuoka University, Faculty of Science, Professor, 理学部, 教授 (80112283)
NISHIMOTO Tetsu Kinki Welfare University, Assosiate Professor, 助教授 (80330520)
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| Project Period (FY) |
2003 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥6,000,000 (Direct Cost: ¥6,000,000)
Fiscal Year 2006: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2005: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2003: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | A_∞-structure / L-S category / Lie group / Hopf invariant / co-Hopf space / A_∞構造 / 特異点 / L-Sカテゴリー / 球面束 / 高次Hopf不変量 / A_∞手法 / L-Sカテゴリ数 / Cone分解 / 主束 / 射影ユニタリー群 |
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| Research Abstract |
L-S category is defined by Lusternik and Schnirelmann to give a homotopy invariant of a topological space, which gives, for a manifold M, a lower bound of the number of the critical points of a C^∞-function on M. After the investigation of Ganea and many others, the following two problems were realised as open problems and listed in the book "Open Problems in Topology" by van Mill and M. Reed.
[Problem 642] (Ganea conjecture) Is the L-S category of a space X × S^n equal to the L-S category of X plus 1?
[Problem 643] Is the L-S category of a closed manifold greater than that of its once punctured submanifold?
The higher Hopf invariant redefined on projective spaces of the loop space of a space, gave a criterion to determine L-S category, which implies negative answers to Problem 642 and Problem 643.
In this research, extensively using the above idea, the present investigator completely determined the L-S category of a manifold which is the total space of a sphere-bundle over a sphere. As a result, we can construct manifolds as total spaces of S^2-bundles, which give counter examples to the Ganea conjecture. With Mamoru Mimura at Okayama University and Tetsu Nishimoto at Kinki Welfare University, he continued to investigate on the total space of a fibre bundle over a simply-connected suspension space or a non-simply-connected space.
Combining this idea with the higher Hopf invariants, he together with Akira Kono at Kyoto University obtained a new upper bound for L-S category. They also defined a new lower bound for L-S category using the A_∞-method. These results yield the determination of L-S category of all compact simple Lie groups up to rank 4,except for Sp(4),F_4,PSp(3) and PSp(4). Using the A_∞-point of view, Kamata, Saeki, Sumi, Oda and Nishimoto have obtained various results.
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