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Finite-dimensional categorial theorems in shape theory. (English) Zbl 0665.55006
Let \({\mathcal S}\) denote the full subcategory of the shape category whose objects are Z-sets in the Hilbert cube Q, and let \({\mathcal W}{\mathcal H}{\mathcal P}\) denote the category whose objects are Z-set complements in Q and whose morphisms are weak proper homotopy classes of proper maps. According to T. A. Chapman’s Categorical Complement Theorem [Fundam. Math. 76, 181-193 (1972; Zbl 0262.55016)] there is a category isomorphism \(\Phi\) : \({\mathcal S}\to {\mathcal W}{\mathcal H}{\mathcal P}\) such that \(\Phi (X)=Q-X\). The author [Fundam. Math. 125, 195-208 (1985; Zbl 0592.57013)] has established a version of this result for Z-embeddings of compacta in AR’s with a complete uniform structure. Well known finite- dimensional complement theorems in Euclidean spaces might lead one to expect a similar result for nice embeddings of compacta in \(R^ n\); of course there are no Z-embeddings in this case. Thus the search for a finite-dimensional categorical complement theorem that will apply in this case centers on finding (a) an appropriate class of ambient spaces and “admissibility embedded” compacta and (b) an appropriate “complementary category” whose objects are complements of admissibly embedded compacta. The author establishes such a theorem. The statement of the theorem involves some technical definitions and shall not be given here. We content ourselves instead with stating one of the several corollaries of the main result: If M is an m-connected ANR, then the shape category of m-admissible compacta \(X\subset M\) with FdX\(\leq m\) is isomorphic to the weak complete m-homotopy category of their complements M-X; if M is compact, one can replace the weak complete m-homotopy category by the weak proper m-homotopy category. Finally, we remark that in the case where M is a piecewise linear manifold, the notion of admissibility can be related to the well-known ILC embedding condition.
Reviewer: R.Sher

MSC:
55P55 Shape theory
57N25 Shapes (aspects of topological manifolds)
54C56 Shape theory in general topology
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