Cartesian powers of shapes of FANR's and polyhedra (Q1676518)

The main result of the paper is: if \(K\) is a topological space homotopy dominated by some polyhedron \(P\) and \(K\simeq K\times L\), then \(L\simeq 1\). In particular, if \(P\) is a polyhedron and \(P^n \simeq P\) (\(n\geq 2\)), then \(P\simeq 1\), and hence \(P\) is an AR. This gives a positive answer to a question posed by \textit{K. Borsuk} in [Theory of shape. Aarhus: Matematisk Institut, Aarhus Universitet (1971; Zbl 0232.55021)]. The author also obtains a similar result for FANR's in shape theory: If \(X\) is an FANR and \(Sh(X) = Sh(X) \times Sh(Y)\), then \(Sh(X)=1\). In particular, if \(X\) is an FANR and \(Sh^n(X) = Sh(X)\) (\(n\geq 2\)), then \(Sh(X)=1\).