The Lattice of Normal Subgroups of the Group of Diffeomorphisms or Homeomorphisms of an Open Manifold
The Lattice of Normal Subgroups of the Group of Diffeomorphisms or Homeomorphisms of an Open Manifold
复制标题
开流形微分同胚或同胚群的正规子群的格
DOI：
10.1112/jlms/s2-18.2.353
复制
发表时间：
1978
影响因子：
1.2
通讯作者：
D. Mcduff
中科院分区：
文献类型：
--
作者：
D. Mcduff
我们将在DifT，r= 0，1，...，oo，其中Diff 0 =定义的顶部。设P是紧致流形P的内部，我们用A（P）表示P的所有具有紧开C-拓扑的自同构（即同同态或同胚）的群，用A0（P）或简称AOi表示A（P）的单位分支。如果SP = 0，除了可能的r= 1+ dim P之外，已知A0（P）是单群。(See[1]、[7]、[8]、[9]、[14]、[19]。另一方面，如果dP # 0，AQ（P）包含非平凡正规子群，例如由A0（P）中具有紧支集的所有自同构组成的子群。然而，我们将证明A0（P）是完美的，它的正规子群可以有效地分类。它们本质上只依赖于P的边界。更精确地说，让我们从现在开始假设P的边界d是非空的，并且它的连通分量是du…，dk。设K为集合{1，.，k}。如果J ∈ K，我们对于M dh写dj，对于子群{me A（P）：m= id near dj}写A（P，rel dj）ieJ。最后，对于每个ie K，设Gt.= A（P，reld;）n Ao，如果J ∈ K，则将
We will work in one of the categories DifT, r= 0, 1,..., oo, where Diff0= Top by definition. Let P be the interior of a compact manifold P. We denote by A (P) the group of all automorphisms (ie diffeomorphisms or homeomorphisms, as the case may be) of P with the compact-open C-topology, and by A0 (P), or simply AOi the identity component of A (P). IfSP= 0 it is known, except possibly if r= 1+ dim P, that A0 (P) is a simple group.(See [1],[7],[8],[9],[14],[19].) On the other hand, ifdP# 0, AQ {P) contains non-trivial normal subgroups, for instance the subgroup consisting of all automorphisms in A0 (P) with compact support. We will, however, show that A0 (P) is perfect and that its normal subgroups can be effectively classified. They depend essentially only on the boundary of P. To be more precise, let us suppose from now on that the boundary d of P is non-empty and that its connected components are du..., dk. Let K be the set {1,..., k}. If J£ K, we write dj for M dh and A (P, rel dj) ieJ for the subgroup {me A (P): m= id near dj}. Finally, for each ie K let Gt.= A (P, reld;) n Ao, and, if J£ K, put