2001 Fiscal Year Final Research Report Summary
Topological study on the structure of the group of homeomorphisms
| Project/Area Number |
12640094
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Kyoto Sangyou University |
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Principal Investigator |
FUKUI Kazuhiko Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (30065883)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Shuji Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (30192404)
USHITAKI Fumihiro Kyoto Sangyo University, Faculty of Science, Associate Professor, 理学部, 助教授 (30232820)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Lipschitz homeomorphism group / 1-dimensional homology / commutator / perfect / foliated manifold / G-manifold / orbifold |
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| Research Abstract |
1. We considered the group of Lipschitz homeomorphisms of a Lipschitz manifold and showed that the group is locally contractible and perfect. As its application, we also showed that the group of equivariant Lipschitz homeomorphisms of a principal G-manifold is perfect when G is a compact Lie group. Furthermore we showed that the group of Lipschitz homeomorphisms of R^n leaving the origin fixed is perfect. As its application, we can show that the groups of Lipschitz homeomorphisms of a Lipschitz orbifold and of foliation preserving Lipschitz homeomorphisms of a compact Hausdorff C^1-foliated manifold are perfect.
2. It is known that the equivariant diffeomorphism group of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. We determined the first homology group of the equivariant diffeomorphism group of M when M is a G-manifold with codimension one orbit.
3. We considered the group of foliation preserving Lipschitz homeomorphisms of a foliated manifold and computed the first homologies of the groups for codimension one C^2-foliations. We showed that if the foliation has no type D components and has only a finite number of type R components, then the group is perfect. Furthermore we showed that if the foliation has a type D component and the linearization map is a C^1-diffeomorphism, then the group is not perfect. But we showed that if the foliation has a type D component and the linearization map is not absolutely continuous, then the group is perfect. This phenomenon is different from that in topological case.
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Research Products
(14 results)
