Properties of normal embeddings concerning strong shape theory. I
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) shape equivalence of pairs.
scientific article; zbMATH DE number 61114