Function spaces and shape theories (Q2778629)

In their endeavour to present the strong shape category for (arbitrary) topological spaces and pro-spaces as a quotient category of an ordinary homotopy category localized along strong shape equivalences, the authors obtain the following particular results: NEWLINENEWLINENEWLINETheorem 1. A map \(f: X \rightarrow Y\) of k-spaces is a strong shape equivalence, if and only if \(f\times id_Q: X \times_k Q \rightarrow Y \times_k Q\) is a shape equivalence for each CW complex \(Q\). NEWLINENEWLINENEWLINETheorem 2. Suppose \(f: X \rightarrow Y\) is a map of topological spaces. (a) \(f\) is a shape equivalence, if and only if the induced function \(f^*: [Y,M] \rightarrow [X,M]\) is a bijection for all \(M=\text{Map}(Q,P)\), where \(P \in {\mathbf {ANR}}\), and \(Q\) is a finite CW complex. (b) If \(f\) is a strong shape equivalence, then the induced function \(f^* : [Y,M] \rightarrow [X,M]\) is a bijection for all \(M= \text{Map}(Q,P)\) where \(P \in{\mathbf {ANR}}\) and \(Q\) is an arbitrary CW complex. (c) If \(X, Y\) are k-spaces and the induced function \(f^*: [Y,M] \rightarrow [X,M]\) is a bijection for all \(M= \text{Map}(Q,P)\) where \(P \in{\mathbf {ANR}}\) and \(Q\) is an arbitrary CW complex, then \(f\) is a strong shape equivalence. NEWLINENEWLINENEWLINEMoreover, the authors want to correct in the present paper some of the errors, which occured in a previous paper. NEWLINENEWLINENEWLINEStrong shape theory was invented independently by D. E. Edwards and H. M. Hastings on one hand and the reviewer about 26 years ago. The present authors (re-)invent the strong shape singular complex (in historical terminology \({\overline S}(X)\)) together with the canonical shape morphism \({\overline \omega}:|{\overline S}(X) |\rightarrow X\), having the same universality properties like the ordinary singular complex of a space, but now in strong shape category (not in the associated homotopy category), cf. for further reference \textit{H. Thiemann} [Cah. Topologie Géom. Différ. Catégoriques 30, No. 2, 157-165 (1989; Zbl 0684.55008)]. The authors conjecture that singular shape homology i.e. the homology of the CW (or alternatively of the simplicial) complex \({\overline S}(X)\), is isomorphic to strong homology, is (even for compacta) since more than 20 years known to be not true [cf. \textit{F. W. Bauer}, Pac. J. Math. 128, No. 1, 25-61 (1987; Zbl 0646.55004), for further references, in particular to \textit{M. Kernchen}, Manuscr. Math. 39, 111-118 (1982; Zbl 0516.55006)]. The point is, that strong homology satisfies a cluster axiom while singular homology admits a universal coefficient sequence. According to Kernchen's result both properties cannot be true together in full generality (i.e. without restrictions on the coefficient spectrum).