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Coherent prohomotopy and strong shape theory. (English) Zbl 0553.55009

The coherent prohomotopy category CPHTop, defined in this paper, has as objects inverse systems \(X=(X_{\lambda},p_{\lambda \lambda '},\Lambda)\) of topological spaces and maps indexed by a directed cofinite index set \(\Lambda\). To define morphisms \(f: X\to Y=(Y_{\mu},q_{\mu \mu '},M)\) one first defines coherent maps of systems \(X\to Y\). These consist first of a function \(\phi\), which associates with each increasing sequence \(m=(m_ 0,...,m_ n)\) in M an element of \(\Lambda\). Moreover, to every m it assigns a map \(f_ m: X_{\phi (m)}\times \Delta^ n\to Y_{\mu_ 0},\) with an appropriate behaviour with respect to the standard boundary and degeneracy operators between standard simplexes \(\Delta^ n\). Morphisms of CPHTop are coherent homotopy classes of coherent maps. A special case of coherent maps is the case when \(\phi (m)=\phi (\mu_ n)\) and \(\phi\) \(| M\) is increasing. Composition of the morphisms in CPHTop is defined by an explicit formula, which shows how to compose special coherent maps representing the given morphisms. The authors believe that the category CPHTop is isomorphic to the homotopy category of inverse systems considered in [D. A. Edwards and H. M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lect. Notes Math. 542 (1976; Zbl 0334.55001)], the advantage consisting in its explicit geometric definition. Using the work of R. M. Vogt [Math. Z. 134, 11-52 (1973; Zbl 0276.55006)], T. Porter has a similar construction in [Cah. Topologie Géom. Différ. 19, 3-46 (1978; Zbl 0387.55013)].
In the second part of the paper a strong shape category SSh is defined having all topological spaces for its objects. This is achieved using the notion of a resolution p: \(X\to \underline{X}\) of a space X [the second author, Fundam. Math. 114, 53-78 (1981; Zbl 0411.54019)]. The crucial theorem relating resolutions and coherent prohomotopy asserts that for any resolution p: \(X\to \underline{X}\) and any morphism \(f: X\to \underline{Y}\) of CPHTop into an ANR-system Y, there exists a unique morphism of CPHTop g: \b{X}\(\to \underline Y\) such that g\b{p}\(=f\). A morphism \(X\to Y\) of SSh is given by two ANR-resolutions p: \(X\to \underline{X}\), q: \(Y\to \underline{Y}\) and by a morphism of CPHTop g: X\(\to \underline{Y}\). Most other approaches to strong shape were restricted to the compact metric case or used coherent prohomotopy only of order \(n=2\). An exception is [F. W. Cathey and J. Segal, Topology Appl. 15, 119-130 (1983; Zbl 0505.55012)], who develop their theory of strong shape using resolutions and the coherent prohomotopy of Edwards and Hastings.

MSC:

55P55 Shape theory
54C56 Shape theory in general topology
55U10 Simplicial sets and complexes in algebraic topology
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