Embedding spaces and hyperspaces of polyhedra in 2-manifolds
type: Item
instance of: scholarly article
title: Embedding spaces and hyperspaces of polyhedra in 2-manifolds
zbMATH Open document ID: 1010.54007
DOI: 10.1016/S0166-8641(01)00121-3
author: Tatsuhiko Yagasaki
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(M\) be a 2-manifold and \(X\) a compact polyhedron. Denote by \({\mathcal E}(X,M)\) the space of embeddings of \(X\) into \(M\) and by \({\mathcal H}(X)\) the group of homeomorphisms of \(X\), both spaces equipped with the compact-open topology. Finally, denote by \({\mathcal K}(X,M)\) the hyperspace consisting of the subsets of \(M\) homeomorphic to \(X\). Equip \({\mathcal K}(X,M)\) with the Fréchet topology, a metric topology defined by considering distances between subsets of \(M\) homeomorphic to \(X\). It is proved that the natural map \(\pi: {\mathcal E}(X,M) \to {\mathcal K}(X,M)\), defined by \({\pi}(f) = f(X)\), is a principal bundle with fibre \({\mathcal H}(X)\). From this the author draws the interesting corollary that \({\mathcal K}(X,M)\) is an ANR.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 55M15
zbMATH DE Number: 1787527
reviewed by: Vagn Lundsgaard Hansen
MaRDI profile type: Publication
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full work available at URL: s0166-8641(01)00121-3
OpenAlex ID: W2093451212
description: scientific article; zbMATH DE number 1787527