CE equivalence and shape equivalence of 1-dimensional compacta
type: Item
instance of: scholarly article
title: CE equivalence and shape equivalence of 1-dimensional compacta
zbMATH Open document ID: 0623.54006
DOI: 10.1016/0166-8641(87)90064-2
author: G. A. Venema
author: Robert J. Daverman
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Finite dimensional compacta X and Y are CE equivalent if there exist a finite dimensional compactum Z and cell-like maps of Z onto both X and Y. While every cell-like map between finite dimensional compacta is a shape equivalence, \textit{S. Ferry} [Proc. Am. Math. Soc. 80, 154-156 (1980; Zbl 0542.57003)] has shown that there is a planar continuum X such that \(Sh(X)=Sh(S^ 1)\) but X is not CE equivalent to \(S^ 1\). The authors study the relationship between CE equivalence and shape equivalence for locally connected 1-dimensional compacta. They show that: (1) every path connected planar continuum is CE equivalent either to a finite bouquet of circles or to the Hawaiian earring; and (2) every locally connected 1- dimensional continuum is the domain f a cell-like map onto a planar continuum. It follows that CE equivalence and shape equivalence agree on the class of all locally connected 1-dimensional compacta. On the other hand, for every \(n\geq 1\) an example is given of an n-dimensional LC\({}^{n-2}\) continuum X in \(R^{n+1}\) such that \(Sh(X)=Sh(S^ 1)\) but X is not CE equivalent to \(S^ 1\).
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54F35
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57N25
zbMATH DE Number: 4011407
zbMATH Keywords: local connectivitiy
zbMATH Keywords: cell-like maps
zbMATH Keywords: CE equivalence
zbMATH Keywords: shape equivalence
zbMATH Keywords: path connected planar continuum
zbMATH Keywords: locally connected 1-dimensional continuum
zbMATH Keywords: locally connected 1-dimensional compacta
zbMATH Keywords: \(LC^{n-2}\) continuum
reviewed by: Richard Sher
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90064-2
OpenAlex ID: W2029151688
cites work: Concerning homotopy properties of compacta
cites work: Some applications of an approximation theorem for inverse limits
cites work: Q3928677
cites work: Homotopy, simple homotopy and compacta
cites work: Shape Equivalence Does Not Imply Ce Equivalence
cites work: Realizing cell-like maps in Euclidean space
cites work: Embeddings of Locally Connected Compacta
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description: scientific article; zbMATH DE number 4011407