The \(n\)-shape of compact pairs and weak proper \(n\)-homotopy (Q2785617)

An analogue of Chapman's category isomorphism theorem [\textit{T. A. Chapman}, Fundam. Math. 76, 181-193 (1972; Zbl 0262.55016)] between the shape category of compact \(Z\)-sets in the Hilbert cube and the category of weak proper homotopy equivalence classes of proper maps of their complements is proved for \(n\)-shape and pairs of compacta. The ambient space for \(n\)-shape and such a statement is the \((n+1)\)-dimensional universal Menger compactum \(\mu^{n+1}\), but for a pair of compacta a \(Z\)-set \(\mu_0^{n+1}\) homeomorphic to \(\mu^{n+1}\) in \(\mu^{n+1}\) is needed. A pair of compacta \((X,X_0)\) 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 author has defined the \(n\)-shape of compact pairs and his main result says that there exists a category isomorphism \(\psi\) from the \(n\)-shape category of \(Z\)-pairs in \((\mu^{n+1}, \mu_0^{n+1})\) and the weak proper \(n\)-homotopy category of complements of \(Z\)-pairs such that \(\psi(X,X_0)= (\mu^{n+1} \smallsetminus X,\mu_0^{n+1} \smallsetminus X_0)\). One may comment that the research was inspired by the \(n\)-shape complement theorem of \textit{A. Chigogidze} [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 paper contains all needed technicalities for compact \(Z\)-pairs in \((\mu^{n+1}, \mu_0^{n+1})\).