Commutators of Foliation Preserving Homeomorphisms for Certain Compact Foliations
Commutators of Foliation Preserving Homeomorphisms for Certain Compact Foliations
复制标题
叶状交换子保持某些紧凑叶状结构的同态
DOI：
10.2977/prims/1195144828
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发表时间：
1998
影响因子：
1.2
通讯作者：
K. Fukui
中科院分区：
文献类型：
--
作者：
K. Fukui
设M是n维闭拓扑流形。用X（M）表示M的所有同胚的群，这些同胚通过在紧集外固定的合痕而与单位元合痕。本文讨论X（M）的某些子群。设（M，N）是流形对，其中A是M的真子流形.设X（M，N）表示X（M）的在N上不变的同胚子群.在§2中，我们考虑了3C（M，A）的同调，即群X（M，N）的同调群，并证明了X（R，R）（p> 0）的同调在> 0的所有维数中为零。这是Fukui-Imanishi [F~ 1]的一个结果的特殊情况，而Fukui-Imanishi [F~ 1]是Mather [Ma]的一个结果在叶状流形上的推广。我们在§3中证明了3C（M，A）是完美的，即，对于一定的流形对（M，N），等于它自己的换位子群。在§4和§5中，我们考虑了保持同胚的叶理群。我们在[F~ 1]中已经讨论了余维为1的叶理的情形。我们在这里研究的情况下，紧叶理的余维大于1。设（M，θ）是C^-叶流形，F（M，θ）是（M，θ）的保叶同胚群，它通过保叶合痕固定在紧流形外而与单位元同胚
Let M be an n-dimensional closed topological manifold. By X (M) we denote the group of all homeomorphisms of M which are isotopic to the identity by an isotopy fixed outside a compact set. In this note we treat certain subgroups of X (M). Let (M, N) be a manifold pair, where A is a proper submanifold of M. Let X (M, N) denote the subgroup of homeomorphisms of X ( M ) which are invariant on N. In §2, we consider the homologies of 3C (M, A), that is, the homology groups of the group X(M, N) and show that the homologies of X(R, R) (p> 0) vanish in all dimension > 0. This is a special case of a result of Fukui-Imanishi [F~l] which is a generalization of a result of Mather [Ma] to foliated manifolds. We show in §3 that 3C (M, A) is perfect, i.e., is equal to its own commutator subgroup, for a certain manifold pair (M, N). In §4 and §5, we consider the group of foliation preserving homeomorphisms. We have already discussed in [F~l] about the case of codimension one foliations. We study here the case of compact foliations of codimension greater than one. Let(M, 9} be a C^-foliated manifold and F(M, &) be the group of foliation preserving homeomorphisms of (M, ^) isotopic to the identity by a foliation preserving isotopy fixed outside a compact