On commutators of foliation preserving Lipschitz homeomorphisms
This article concerns the group of foliation preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold. Let \(M\) be an \(m\)-dimensional, connected, Lipschitz manifold, \({\mathcal F}_0\) the \(p\)-dimensional foliation of \(\mathbb{R}^m\) whose leaves are defined by \(x_{p+1}= \text{const},\dots, x_m=\text{const}\) \((1\leq p\leq m)\). By definition, a \(p\)-dimensional Lipschitz foliation \({\mathcal F}\) of \(M\) is a maximal set of local Lipschitz charts \(\{(U_\alpha, \varphi_\alpha)\}_{\alpha\in A}\) such that \(\varphi_\alpha\circ \varphi^{-1}_\beta\) preserves the leaves of the foliations \({\mathcal F}_0|\varphi_\beta(U_\alpha\cap U_\beta)\) and \({\mathcal F}_0|\varphi_\alpha(U_\alpha\cap U_\beta)\). A Lipschitz homeomorphism \(f: M\to M\) is said to be leaf preserving, if for each point \(x\in M\), \(f(L_x)= L_x\), where \(L_x\) is the leaf of \({\mathcal F}\) that contains \(x\). Let \({\mathcal H}_{LIP,L}(M,{\mathcal F})\) denote the group of the leaf preserving maps that are isotopic to the identity by the same kind of Lipschitz homeomorphisms fixed outside a compact set. Foliation preserving homeomorphism is defined in the same way except for ``\(f(L_x)= L_{f(x)}\)'', and the corresponding space \({\mathcal H}_{LIP}(M,{\mathcal F})\) of homeomorphisms is introduced. In the first part of the paper, the authors show that the homologies of \({\mathcal H}_{LIP,L}(\mathbb{R}^m,{\mathcal F}_0)\) vanish in all dimensions \(>0\). Next, a proposition called fragmentation lemma is derived under the assumption that \(M\) is compact. That is, every element of \({\mathcal H}_{LIP,L}(M,{\mathcal F})\) is expressed as a product of leaf preserving maps with support in small balls. It follows from the first result above that \({\mathcal H}_{LIP,L}(M,{\mathcal F})\) is perfect, provided that \(M\) is compact. Local contractibility of \({\mathcal H}_{LIP,L}(M,{\mathcal F})\) is immediately obtained, too. In the final section, the first homology of \({\mathcal H}_{LIP}(M,{\mathcal F})\) for a codimension 1, \(C^2\) foliated manifold \((M,{\mathcal F})\) is developed. If \({\mathcal F}\) has no dense components (with an additional assumption), \({\mathcal H}_{LIP}(M,{\mathcal F})\) is perfect, which is the same as that in the topological case but is different from that in the differentiable case. While if \({\mathcal F}\) has a dense component, a different situation arises from that in the topological case.
scientific article; zbMATH DE number 1803230