The autornorphisrn group of smcoth G-manifokls andals applications

设M是具有紧李群G的光滑作用的光滑流形，Lip_G(M)表示M的等变Lipschitz同胚群，我们可以在Lip_G(M)上引入两种拓扑。设Lip_G(M)，(Rep.L_G(M)表示当Lip_G(M)具有紧开Lipschitz拓扑时，Lip_G(M)的恒等式的单位分支。紧凑开放拓扑)。研究了H_i(Lip_G(M)_O)的第一同调群，当M是主G-流形或G是有限群时，我们知道Lip_G(M)_O群是完全群。当M是具有标准U(N)作用的n维复空间C^n时，H_1(L_G(M))同构于一些具有连续模的实值函数的向量空间.在研究期间，我们研究了M有余维为一个轨道的G-流形的情况.首先证明了当V是余维单轨表示空间时，Lip_G(V)_o是完全的。利用这一结果，通过分析奇异轨道附近等变Lipschitz同胚的性质，我们计算了任意余维为一个轨道的光滑G流形的H_1(Lip_G(M)-O)。结果表明，光滑G流形的自同构群的第一同调群与光滑、紧开或紧开的Lipschitz范畴密切相关。还计算了三维光滑S流形的等变微分同胚群的第一同调群。这项研究结果最近发表在《外国杂志》上。

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.