Infinite-dimensional manifold triples of homeomorphism groups
An \((l+1)\)-tuple \((X,X_1,X_2,\dots, X_l)\) is called an \((E,E_1,E_2,\dots, E_l)\)-manifold if each point of \(X\) has an open neighbourhood \(U\) and an open embedding \(h:U\to E\) such that \(h(U\cap X_1)= h(U)\cap E_1,\dots, h(U\cap X_l)= h(U)\cap E_l\). In the present paper the author considers and gives a characterization of \((S,S_1,\dots, S_l)\)-manifolds, where \(S=(-\infty,\infty)^\infty\) and \(S_1,\dots, S_l\) are linear subspaces of \(S\). Then he proves the conjecture about the homeomorphism group triple, established by \textit{K. Sakai} and \textit{R. Y. Wong} [Pac. J. Math. 139, No. 1, 195-207 (1989; Zbl 0677.57011)] and that if \(X\) is a compact Euclidean polyhedron with \(\dim X=1\) or 2, then the triple \((H(X), H^{\text{LIP}} (X), H^{\text{PL}} (X)))\) is an \((S,\Sigma, \sigma)\)-manifold (\(H(X)\) is the homeomorphism group of \(X\) with the compact-open topology; \(H^{\text{Lip}} (X)\), and \(H^{\text{PL}} (X)\) are the subgroups of \(H(X)\) consisting of Lipschitz homeomorphisms and PL-homeomorphisms, respectively; \(\Sigma= \{(x_n)\in S\mid\sup_n|x_n|<\infty\}\) and \(\sigma= \{(x_n)\in S\mid x_n=0\) except for finitely many \(n\geq 1\}\).
scientific article; zbMATH DE number 1022181
scientific article