Products of hopfian manifolds and codimension-2 fibrators (Q1566932)

A closed orientable manifold is hopfian if every degree one self-map is a homotopy equivalence. A codimension \(2\) fibrator is a closed \(n\)-manifold \(N\) with the property that whenever \(p:M\to X\) is a map from a closed \((n+2)\)-manifold onto a space \(X\) with each point inverse \(p^{-1}(x)\) shape equivalent to \(N\), then \(p\) is an approximate fibration. \textit{R. J. Daverman} [Compos. Math. 86, No. 2, 159-176 (1993; Zbl 0788.57012)] has established interesting relations among hopfian manifolds, fibrators and the hopfian and hyperhopfian properties of groups. The authors describe several conditions under which the product of closed, orientable manifolds \(F\) and \(A\) is hopfian or a codimension \(2\) fibrator. For example, if the fundamental group of \(F\) is finite and \(A\) is aspherical, then \(F\times A\) is hopfian. If, in addition, the fundamental groups of \(F\) and \(A\) are hyperhopfian (i.e., every self-homomorphism with normal image and cyclic co-image is an isomorphism), then \(F\times A\) is a codimension \(2\) fibrator.