1999 Fiscal Year Final Research Report Summary
Study on the structure of the group of homeomorphisms
| Project/Area Number |
10640096
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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 Sangyo University |
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Principal Investigator |
FUKUI Kazuhiro 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, Associate Professor, 理学部, 助教授 (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) |
1998 – 1999
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| Keywords | foliated manifold / homeomorphism group / commutator / homology / Lipschitz homeomorphism group / G-manifold / compact. leaf / stable |
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| Research Abstract |
1. We considered the group of foliation preserving homeomorphisms of a foliated manifold and computed the first homologies of the groups for condimension one 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. Especially the group for the Reeb foliation on the 3-sphere is perfect. Furthermore we showed than if the foliation preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold and computed the first homology of the group of foliation preserving Lipschitz homeomorphisms of a codimension one CィイD11ィエD1-foliated manifold. Then we have a phenomenon different from that in topological case.
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 stability of compact leaves. Then we showed that all compact Hausdorff CィイD1rィエD1-foliations of 4-manifolds by hyperbolic surfaces are not stable (1 【less than or equal】 r【less than or equal】∞).
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