Commutators of diffeomorphisms
Commutators of diffeomorphisms
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DOI：
10.1007/bf02566746
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发表时间：
1974-12
影响因子：
0.9
通讯作者：
J. Mather
中科院分区：
文献类型：
--
作者：
J. Mather
在本文中，我们将证明某些微分同态群是完全的，即等于它们自己的对易子群。Epstein已经证明I-2]对于相当一般的同胚群，交换子群是简单的。特别地，他的结果表明对于我们所考虑的微分同态群，交换子群是简单的。将他的结果与我们的结果结合起来，我们发现我们所考虑的群体是简单的。在w7中，我们得到了一个关于叶的Haefliger分类空间的连通性的结果，作为我们的证明的一个推论和Thurston[4]的结果。我们说空间M中的同位素Ht有紧化支持，如果M中有一个紧化集合K使得Ht (x)= x对于所有的xeM-K和所有的t。设M是一个光滑流形。我们将Diff （M, r）定义为M的C - r微同形的一组，这些微同形通过紧支C - r同位素同形。因此，Diff （M, r）的任何元素都有紧支撑。我们的主要结果如下。
In this paper, we will show that certain groups of diffeomorphisms are perfect, ie, equal to their own commutator subgroups. Epstein has shown I-2] that for quite general groups of homeomorphisms, the commutator subgroup is simple. In particular, his result shows that for the groups of diffeomorphisms which we consider, the commutator subgroup is simple. Combining his results with our result, we see that the groups we consider are simple. In w 7, we obtain a result concerning the connectivity of Haefliger's classifying space for foliations as a corollary of our proof and a result of Thurston [4].We say an isotopy Ht of a space M has compact support if there is a compact set K in M such that Ht (x)= x for all xeM-K and all t. Let M be a smooth manifold. We define Diff (M, r) to be the group of C r diffeomorphisms of M which are isotopic to the identity through compactly supported C r isotopies. Thus, any element of Diff (M, r) has compact support. Our main result is the following.