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Finite-dimensional categorical complement theorems in strong shape theory and a principle of reversing maps between open subsets of spheres. (English) Zbl 0721.55006
This paper continues the author’s work in finding generalized shape complement theorems. In a previous paper [Compos. Math. 68, 161–173 (1988; Zbl 0665.55006)] the author proved a theorem which related the shape of a space \(X\) embedded in an ANR \(M\) to homotopical information about the complement of \(X\) in \(M\). The theorem requires restrictions on the embedding, the fundamental dimension of \(X\), and the connectivity of \(M\). The present paper is concerned with relating the strong shape of \(X\) to the complement of \(X\) embedded in an appropriate way in an appropriate ANR. The theorems obtained bear an appropriate resemblance of those obtained in his previous paper. The following duality theorem in the paper is quoted to give a flavor of the results obtained.
Theorem B. Let \(d(n)=\max \{k\mid 2k+2\leq n\}\) and \(c(n)=\max \{k\mid 4k\leq n\}\). Let \(T_ n\) be the full subcategory of the homotopy category HTop whose objects are all complements \(S^ n\setminus X\) of \(c(n)\)-shape-connected compacta \(X\subset S^ n\) having \(\text{Fd }X\leq d(n)-1\) and satisfying the inessential loops condition ILC, and let \(H_{d(n)}P\) be the proper \(d(n)\)-homotopy category. There exists a contravariant full embedding \(\Delta : T_ n\to H_{d(n)}P\) such that \(\Delta (U)=U\) for each object \(U\).
The results of the author are more categorical to state and prove than the shape complement theorems of T. Chapman [Fundam. Math. 76, 181–193 (1972; Zbl 0262.55016) and ibid. 76, 261–276 (1972; Zbl 0222.55019)]. Other attempts as generalization also have had a more geometrical flavor [see R. B. Sher, Lect. Notes Math. 870, 150–168 (1981; Zbl 0494.57007) and ibid. 1283, 212–220 (1987; Zbl 0631.55006)].

MSC:
55P55 Shape theory
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References:
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