CE equivalence and shape equivalence of LC\(^ n\) compacta
type: Item
instance of: scholarly article
title: CE equivalence and shape equivalence of LC\(^ n\) compacta
zbMATH Open document ID: 0788.55007
DOI: 10.1016/0166-8641(93)90069-P
author: Peter Mrozik
published in: Topology and its Applications
publication date: 1993-09-22T00:00:00Z
review text: The author continues to investigate the relationship of shape equivalence and \(UV^ m\) equivalence of path connected compacta. It is known that \(UV^ m\)-equivalent \(m\)-dimensional compacta are shape equivalent, while shape equivalent \(m\)-dimensional compacta need not be \(UV^ m\) equivalent. S. Ferry has proved the following: If \(X\) is a continuum with pro-\(\pi_ 1X\) profinite and \(Y\) shape equivalent to \(X\), then \(Y\) is \(UV^ m\) equivalent to \(X\) for each \(m \geq 2\). [\textit{S. C. Ferry}, `Geometric topology and shape theory', Lect. Notes Math. 1283, 88-114 (1987; Zbl 0634.55008)]. The main result of reviewed paper is: If \(X\) is an \(LC^{n+1}\) continuum, \(n\geq 1\), with \(\pi_ 1(X)\) infinite, then there exists an \(LC^ n\) continuum \(X'\), \(\dim X'\geq \max\{\dim X,n+2\}\) such that \(X\) and \(X'\) are shape equivalent but not \(UV^{n+1}\) equivalent. In particular, \(X\) and \(X'\) are not \(CE\) equivalent. Ferry's first constructed example of shape equivalent compacta which are not \(CE\) equivalent (they are not even \(UV^ 1\) equivalent) falls under this theorem. The core of the paper occupies the founding of \(UV^ m\) and \(CE\) invariants that are later used to detect \(UV^ m\) and \(CE\) inequivalence, respectively. For a pointed space \((X,x_ 0)\), these invariants are named \(UV^ m\) groups and \(CE\) groups, denoted \(\pi^{(m)}_ k(X,x_ 0)\) and \(\pi^{CE}_ k(X,x_ 0)\), \(k = 1,2,\dots\). The \(UV^ m\) groups turned out to be trivial for \(k > m\) and any space \(X\), and isomorphic to the \(k\)-th shape group \(\check\pi_ k(X,x_ 0)\) when \(X\) is an \(LC^ n\) continuum and \(k\leq n,m\). They are formally introduced as certain equivalence classes of triples \((C,\alpha,\beta)\) where \(C\) is \(UV^ m\) or \(CE\) compactum, respectively, \(\alpha: S^{k-1}\to C\) and \(\beta: C\to X\) maps \(S^{k- 1}\) is the \((k-1)\)-sphere and \(\alpha\beta(S^{k-1}) = \{x_ 0\}\). Intuitively, it is a generalisation of elements of \(\pi_ k(…
reviewed by: Ivan Ivanšić
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55Q07
zbMATH DE Number: 416864
zbMATH Keywords: CE equivalence
zbMATH Keywords: \(LC^ n\) compacta
zbMATH Keywords: \(UV^ k\) groups
zbMATH Keywords: shape equivalence
zbMATH Keywords: \(UV^ m\) equivalence
MaRDI profile type: Publication
cites work: Characterizing 𝑘-dimensional universal Menger compacta
cites work: Q5532996
cites work: Homotopical approach to strong shape or completion theory
cites work: On \(UV^ n\)-equivalent compacta
cites work: CE equivalence and shape equivalence of 1-dimensional compacta
cites work: Shape Equivalence Does Not Imply Ce Equivalence
cites work: A Stable Converse to the Vietoris-Smale Theorem with Applications to Shape Theory
cites work: Homotopy, simple homotopy and compacta
cites work: Q3772910
cites work: Q4776620
cites work: Local behavior and the Vietoris and Whitehead theorems in shape theory
cites work: Q3959112
cites work: Some new algebraic methods in topology
cites work: Continua that are shape-equivalent but not \(UV^ 1\)-equivalent
cites work: Mapping Cylinders of Approaching Maps and Strong Shape
cites work: Q5651883
cites work: Covering spaces of paracompact spaces
description: scientific article; zbMATH DE number 416864