2007 Fiscal Year Final Research Report Summary
The autornorphisrn group of smcoth G-manifokls andals applications
| Project/Area Number |
18540077
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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 | Shinshu University |
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Principal Investigator |
ABE Kojun Shinshu University, Faculty of Science, Associate Professor (30021231)
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| Project Period (FY) |
2006 – 2007
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| Keywords | diffeornorphisrn group / smooth G-manifold / smooth orbifold / first homology group / Lipshcitz homeomorphism group |
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| Research Abstract |
Let M be a smooth manifold which has a smooth action of a compact Lie group G. Let Lip_G(M) denote the group of equivariant Lipschitz homeomorphisms of M. We can introduce two kind of topologies on Lip_G(M). Let Lip_G(M), (resp. L_G(M)) denote the identity component of the identity of Lip_G(M) when Lip_G(M) has the compact open Lipschitz topology (resp. compact open topology). We investigate the first homology group of H_I(Lip_G(M)_O).
When M is a principal G-manifold or G is a finite group, it is known that the group LiP_G(M)_O is perfect. When M is the complex n-dimensional space C^n with the canonical U(n)-action,H_1(L_G(M)) is isomorphic to the vector space of some real valued functions which has continuous moduli.
During the study duration, we investigate the case where M has a G-manifold with codimension one orbit. First we proved that Lip_G(V)_o is perfect when V is a representation space with codimension one orbit. By using this result and analyzing the behavior of equivariant Lipschitz homeomorphism around the singular orbit, we calculate H_1(Lip_G(M)-O) for any smooth Gmanifold with codimension one orbit. The result shows that the first homology group of the automorphism group of smooth Gmanifold is quite depends on the category such as smooth, compact open or compact open Lipschitz. We also calculate the first homology group of the group of equivariant diffeomorphisms of 3-dimensional smooth S^Imanifold. The result has recently published in the foreign journal.
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Research Products
(16 results)
