On the first homology of the group of equivariant Lipschitz homeomorphisms (Q819556)

Let \(G\) be a compact Lie group. Let \(L_G(M)\) denote the group of equivariant Lipschitz homeomorphisms of a smooth \(G\)-manifold \(M\) which are isotopic to the identity through equivariant Lipschitz homeomorphisms with compact support. This paper concerns the first homology of the group \(L_G(M)\) which is defined as the quotient of \(L_G(M)\) by its commutator subgroup. The first result in this paper is to show that \(L_G(M)\) is perfect if \(M\) is a smooth principal \(G\)-manifold. Though the result is the same as that for the subgroup \({\mathcal H}_{L,I,P,G} (M)\) which was observed before by the same authors, the present proof is quite different from what it was at that time. Secondly, they consider the case of \(\mathbb{C}^n\) with the canonical \(U(n)\)-action, and they prove that the group \(L_{U(n)} (\mathbb{C}^n)\) is not perfect by showing that the first homology group admits continuous moduli. Finally, applying the above results, they prove that the group \(L(\mathbb{C},0)\) of Lipschitz homeomorphisms of \(\mathbb{C}\) which are isotopic to the identity through compactly supported Lipschitz homeomorphisms fixing the origin also admits continuous moduli.