Infinite-dimensional manifold triples of homeomorphism groups
type: Item
instance of: scholarly article
title: Infinite-dimensional manifold triples of homeomorphism groups
zbMATH Open document ID: 0884.57019
DOI: 10.1016/S0166-8641(96)00104-6
author: Tatsuhiko Yagasaki
published in: Topology and its Applications
publication date: 1998-04-02T00:00:00Z
review text: 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 \(n\geq 1\}\).
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 1022181
zbMATH Keywords: infinite-dimensional manifold
zbMATH Keywords: strong universality
zbMATH Keywords: homeomorphism group triple
zbMATH Keywords: \((S,\Sigma, \sigma)\)-manifold
reviewed by: Enrique Outerelo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1022181