The complement theorem in \(n\)-shape theory for compact pairs (Q2785618)

An analogue of Chapman's complement theorem [\textit{T. A. Chapman}, Fundam. Math. 76, 181-193 (1972; Zbl 0262.55016)] has been proved by \textit{A. Chigogidze} for \(n\)-shape using the universal Menger space \(\mu^{n+1}\) as the ambient space [Math. USSR Sb. 61, No. 2, 471-484 (1988); translation from Mat. Sb., Nov. Ser. 133(175), No. 4(8), 481-496 (1987; Zbl 0669.54010)]. The authors are extending Chigogidze's result to pairs of compacta. Let \(\mu_0^{n+1}\) be a \(Z\)-set homeomorphic to \(\mu^{n+1}\) in \(\mu^{n+1}\). \((X,X_0)\subset (\mu^{n+1}, \mu_0^{n+1})\) is called a \(Z\)-pair in \((\mu^{n+1}, \mu_0^{n+1})\) if \(X\) and \(X_0\) are \(Z\)-sets in \(\mu^{n+1}\) and \(\mu_0^{n+1}\) respectively, and \(X_0=X\cap \mu_0^{n+1}\). The complement theorem then reads as follows: For any two \(Z\)-pairs of compacta \((X,X_0)\) and \((Y,Y_0)\) in \((\mu^{n+1}, \mu_0^{n+1})\), the \(n\)-shape of \((X,X_0)\) equals the \(n\)-shape of \((Y,Y_0)\) iff their complements are homeomorphic i.e. \((\mu^{n+1} \smallsetminus X,\mu_0^{n+1} \smallsetminus X_0)\approx (\mu^{n+1} \smallsetminus Y,\mu_0^{n+1} \smallsetminus Y_0)\). The first named author has introduced the \(n\)-shape of pairs of compact in [ibid., 295-306 (1996; Zbl 0877.55002), see the review above]. In order to prove the quoted complement theorem some further knowledge about \(\mu^{n+1}\)-manifolds was needed. Therefore the paper also contains related results as \(Z\)-pair approximation theorem and \(Z\)-pair unknotting theorem. The absolute case of approximation and unknotting theorems have been proved [in: \textit{M. Bestvina}, Characterizing \(k\)-dimensional universal Menger compacta, Mem. Am. Math. Soc. 71, No. 380 (1988; Zbl 0645.54029)].