Properties of normal embeddings concerning strong shape theory. I
type: Item
instance of: scholarly article
title: Properties of normal embeddings concerning strong shape theory. I
zbMATH Open document ID: 0758.54005
author: Bernd Günther
published in: Tsukuba Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: An open cover of a topological space is called normal if it admits a subordinated, locally finite partition of unity. The author isolates a key condition for a subspace \(A\) of a space \(X\). Namely, \(A\) is said to be normally embedded if for every normal cover \(\mathcal U\) of \(A\), there is a normal cover of \(X\) whose trace on \(A\) refines \(\mathcal U\). He then characterizes normally embedded subspaces \(A\) in terms of approximate extensions \(g\) of maps \(f\) of \(A\) to ANR's \(P\), where the map \(g\) is defined on what is called a normal neighborhood of \(A\) and so that \(g\) approximates \(f\) on \(A\) up to an arbitrary given open cover of \(P\). The principal objective of this work is to use the notion of normal embedding to extend concepts of strong shape theory [\textit{J. Dydak} and \textit{S. Nowak}, Trans. Am. Math. Soc. 323, 765-796 (1991; Zbl 0754.55009), \textit{S. Mardešić} and \textit{J. Segal}, ``Shape Theory'' (1982; Zbl 0495.55001)]. A useful fact, Corollary 2.5, is that whenever \(A\) is normally embedded in \(X\), then the normal neighborhoods of \(A\) form what is called a strong expansion of \(A\). Assume that \(A\), \(B\) are closed in \(X\), \(X=A\cup B\), and that \(A\cap B\) is normally embedded in \(X\). The author then has three theorems which he refers to as `` Mayer- Vietoris'' types: (1) The triad \((X;A,B)\) is excisive with respect to any homology or cohomology functor factoring over the strong shape category. (2) The shape dimension satisfies \(\text{sd } X\leq\max\{\text{sd }A,\;\text{sd }B,\;1+\text{sd}(A\cap B)\}\). (3) If \(Y\) is an arbitrary space and \(\alpha: A\to Y\), \(\beta: B\to Y\) are strong shape morphisms which agree on \(A\cap B\), then there is a strong shape morphism \(\gamma: X\to Y\) which extends both \(\alpha\) and \(\beta\). A fourth theorem states that if \(A\) is normally embedded in \(X\), then \(i: A\to X\) is a strong shape equivalence if and only if \(j: (A,A)\to (X,A)\) is an (ordinary…
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 55N40
zbMATH DE Number: 61114
zbMATH Keywords: normal neighborhood
zbMATH Keywords: normal embedding
zbMATH Keywords: strong shape
zbMATH Keywords: strong expansion
zbMATH Keywords: Mayer-Vietoris
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496161655
OpenAlex ID: W1598349439
DOI: 10.21099/TKBJM/1496161655
description: scientific article; zbMATH DE number 61114