Generalized stable shape and duality
In a previous paper the author and \textit{J. Segal} [Topology Appl. 63, No. 2, 139-164 (1995; Zbl 0831.55010)] introduced a generalized stable shape category \({\mathbf {Sh}}_{\text{spec}}\), containing topological spaces and CW-spectra as objects. The main result of the present paper is the following theorem: (1) For each compact metric space \(X\), there exists a CW-spectrum \(X^\ast\) and a natural isomorphism \(\tau: {\mathbf {Sh}}_{\text{spec}}(Y \wedge X, E) \rightarrow {\mathbf {Sh}}_{\text{spec}} (Y, X^\ast \wedge E)\), for any compact Hausdorff \(Y\) and CW-spectrum \(E\). This \((\cdot)^*\) assignment satisfies a certain (expected) functoriality property. There is a coshape category for spectra \({\mathbf {coSh}}_{\text{spec}}\) introduced, such that the following assertion holds: (2) For any compact metric spaces \(X\) and \(Y\) there exists an isomorphism \[ D:{\mathbf {Sh}}_{\text{spec}}(X,Y) \rightarrow {\mathbf {coSh}}_{\text{spec}}(Y^\ast , X^\ast). \] Result (1) is offered as a generalization of classical S-duality to more general spaces. In particular, the author emphasizes that his spaces \(X\) have not to be finite dimensional. As his predecessors he mentions earlier results by \textit{E. Lima} (1959) and \textit{H. W. Henn} (1981), ignoring the fact that there exists since several years a full S-duality for arbitrary subsets of an \(n\)-sphere by using {compact open strong shape} (so-called {coss-shape}) by the reviewer (having appeared in the same journal as the present paper and in Fundamenta). On the other hand, the author's main theorem does {not} describe a duality but only a kind of adjointness: The main property for a duality would be a natural isomorphism \((X^\ast)^\ast \approx X\) which, apart from the fact that \(E^\ast\) for a spectrum is not defined, can not be expected for objects of this generality, according to well-known basic counterexamples by J. M. Boardman and T. Y. Lin.
scientific article; zbMATH DE number 1566035
scientific article