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，设Lip_G(M)表示M的等变Lipschitz同胚群，在Lip_G(M)上可以引入两种拓扑。设Lip_G(M), (p。L_G(M))表示当Lip_G(M)具有紧致开Lipschitz拓扑时，Lip_G(M)的恒等式分量。紧凑开放拓扑)。我们研究了H_I（Lip_G(M)_O）的第一个同源群。当M是主G流形或G是有限群时，已知群LiP_G(M)_O是完全群。当M是具有正则U(n)作用的复n维空间C^n时，H_1（L_G(M)）同构于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具有余维为一轨道的g流形的情况。首先证明了当V是余维为1轨道的表示空间时，Lip_G(V)_o是完全的。利用这一结果，分析了奇异轨道周围等变Lipschitz同胚的性质，我们计算了任意余维为1轨道的光滑g流形的H_1（Lip_G(M)-O）。结果表明光滑g流形自同构群的第一同构群很大程度上依赖于光滑、紧开或紧开Lipschitz等范畴。我们还计算了三维光滑S^ imanfold的等变微分同态群的第一个同调群。这一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.