A relation between \(k\)-th \(UV^{k+1}\) groups and \(k\)-th strong shape groups (Q1345351)

For a continuum \(X\), \(\pi^{(n)}_k (X)\), \(\underline{\pi}^{CE}_k (X)\) and \({\mathbf \pi}_k(X)\) denotes the \(k\)-th \(UV^n\)-homotopy group, the \(k\)-th CE-homotopy group and the \(k\)-th strong shape group of \(X\). \textit{G. A. Venema} has proved in [Topology Appl. 50, 35-46 (1993; Zbl 0788.55008)] that for every continuum \(X\) it holds \(\pi^{(k+1)}_k (X) = \pi^{(k+2)}_k(X) = \dots \pi^{CE}_K (X)\), and if \(X\) is \(UV^k\)-continuum then \(\pi^{(k)}_k (X) = 0\). The article under review puts in relation \(\pi^{(k+1)}_k(X)\) and \(\underline{\pi}_k(X)\). The main result says the following: There is a homomorphism \(s_k : \underline{\pi}_k^{(k+1)}(X,x_0) \to \underline{\pi}_k (X,x_0)\) and if \(\text{pro-}\pi_1(X)\) is profinite then \(s_k\) is an isomorphism. As a consequence one has that a continuum \(X\) having \(\text{pro-}\pi_1(X)\) profinite is an \(UV^n\)-continuum if \(\pi^{(1)}_0 (X) = \{X\}\) (\(X\) is \(UV^{-1}\)-connected) and for each \(x_0 \in X\) holds \(\pi^{(k+1)}_k (X) = 0\), \(k = 1,2,\dots, n\).