A strong shape theory admitting an \(S\)-dual (Q1892173)

The Spanier-Whitehead duality theorem of \(S\)-theory says that for subpolyhedra \(X,Y\subset S^n\), \(\{X,Y\}\approx\{S^n-Y,S^n-X\}\), where \(\{X,Y\}\) denotes the group of \(S\)-maps, and such an isomorphism does not hold for more general subsets of \(S^n\). \textit{E. L. Lima} [Summa Brasil. Math. 4, 91-147 (1959; Zbl 0102.38304)] has defined new groups in terms of which the Spanier-Whitehead duality theorem is established for arbitrary compacta \(X, Y \subset S^n\). The paper under review establishes an \(S\)-dual for separable metrizable finite dimensional spaces. As the first a lacuna in [the author, Topology Appl. 40, 17-21 (1991; Zbl 0724.55005)] is corrected so that for compacta \(X\) and \(Y\) in \(S^n\) the group of stable strong shape morphisms \(\overline{\{X,Y\}}\) is with respect to inclusions naturally isomorphic to the group of \(S\)-maps \(\{\Sigma(S^n-Y),\Sigma(S^n-X)\}\). Note that the \(n\)-dual of \(X\) here equals \(D_nX=\Sigma(S^n-X)\) and to ensure the metrizability, the symbol \(\Sigma(S^n-X)\) denotes the (unreduced) suspension set of \(S^n-X\) with the appropriate topology which does not change the homotopy type of suspension. In order to deal with noncompact separable metric space embeddable in some \(S^n\) the author introduces the concept of compact-open strong shape (coss) or shape with compact carrier. By omitting a description of sets of morphisms of coss, denoted \(\{X, Y\}_c\), too long to describe here, which is given in terms of stable strong shape morphisms for compacta, it is shown that coss is a stable category having metrizable separable spaces as objects, which has the stable strong shape category of compacta as its full subcategory (if \(X,Y \subset S^n\) are compacta, then \(\{X, Y\}_c \approx \overline{\{X,Y\}}\)). Furthermore, the above result for compacta becomes a corollary of the general case. A result that resembles to Spanier-Whitehead duality theorem says the following: If \(X, Y \subset S^n\), then there exists an isomorphism \(\phi:\{X,Y\}_c \to \{\Sigma (S^n-Y), \Sigma(S^n-X)\}_c\) of Abelian groups which is natural with respect to any coss-morphism. Furthermore, \(\phi(1_X)=1_X\) and \(\phi(\beta \alpha)=\phi(\alpha) \phi(\beta)\), whenever \(\beta \alpha\) is defined. Using all developed results needed for just quoted theorem an \(S\)-category \(\mathcal S\) in the sense of [\textit{J. M. Cohen}, Stable homotopy, Lect. Notes Math. 165, 72-74 (1970; Zbl 0201.55703)]is introduced. The objects of \(\mathcal S\) are pairs \({\mathbf X}=(X,m)\) with \(X\) a finite-dimensional separable metrizable space and \(m \in {\mathbb{Z}}\) integer. A morphism \(\underline{\alpha}:(X,m) \to (Y,n)\) of \(\mathcal S\) is determined by a coss-morphism \(\alpha:\Sigma^k X \to \Sigma^l Y\) where \(k+m=l+n\). After introducing a duality operator \(D:{\mathcal S} \to {\mathcal S}\) its following properties are proved: (1) \(D^2 {\mathbf X} \approx {\mathbf X}\); (2) There is an isomorphism of Abelian groups \(\phi:\{{\mathbf X},{\mathbf Y}\}_c \approx \{D{\mathbf Y}, D{\mathbf X}\}_c\), which also has certain naturality property. For the purpose of better understanding of coss-morphisms, an example is added which analyzes coss-equivalence between \(\Sigma(\Sigma(S^n - X))\) and \(\Sigma(S^{n+1} - X)\).