Commutators of Foliation Preserving Homeomorphisms for Certain Compact Foliations

Abstract

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