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(2010 / vol.12 / no.1)
On the first homology of the automorphism groups of G-manifolds
K=ojun Abe
Pages. 91-102   



We have been calculated the first homology group of the equivarinat
diffeomorphism, the equivariant Lipschitz homeomorphism group and
the equivariant homeomorphism group of $G$-manifolds.
Those first homology groups are
quite different by the category of the automorphism group and reflect
the feature of the category.
We shall describe the results and give the outline of the proof.



0. Preliminaries
1. Diffeomorphism groups
2. Equivariant diffeomorphism groups
3. Equivariant Lipschitz homeomorphism groups
4. Equivariant homeomorphism groups
5. Examples
6. $H_1(D_{U(1)}(C))$
7. ${mathcal H}_{LIP,U(1)}(C)$
8. $H_1(L_{U(1)}(C))$
9. ${mathcal H}_{U(1)}(C)$



$G$-manifold;equivariant diffeomorphism group;first homology group;Lipschitz homeomorphism group;homeomorphism group;codimension one orbit



58D05; 58D10; 57S05