Coperfectly Hopfian groups and shape fibrator's properties (Q2324554)

There is a series of results that investigate proper PL approximate fibration maps \(p:M^{n+k}\to B\) having a closed manifold \(N^n\) as the fiber. A natural question is: which manifold \(N\) may be a fiber, forcing \(p\) to be an approximate fibration. The same author in a previous paper [Topology Appl. 153, No. 15, 2765--2781 (2006; Zbl 1099.57021)] introduced a closed orientable \(n\)-manifold \(N\), called codimension-\(k\)-shape \(\mathrm{m}_{\text{simpl}}\) fibrator that is forcing a PL map \(p:M\to B\) from any closed orientable \((n+k)\)-manifold \(M\) to a triangulated manifold \(B\), such that each point inverse has the same homotopy type as \(N\), to be an approximate fibration. The paper under review applies the concept of \(\mathrm{m}_{\text{simpl}}\) fibrator to the product of manifolds and identifies it among direct products of Hopfian manifolds. Among algebraic preparations the author introduces a class of coperfectly Hopfian groups (such are e.g. all finitely generated Abelian groups). All this is prepared for treating direct products and the obtained result says: The direct product of two closed orientable manifolds homotopically determined by \(\pi_1\) and with coperfectly Hopfian fundamental groups is a shape \(\mathrm{m}_{\text{simpl}}\)o-fibrator if it is a Hopfian manifold and a codimension-2 shape \(\mathrm{m}_{\text{simpl}}\)o-fibrator.