Manifolds with nonzero Euler characteristic and codimension-2 fibrators

A closed connected n-manifold N is called a codimension-2 fibrator (codimension-2 orientable fibrator, respectively) if each proper map p : M → B on an (orientable, respectively) (n + 2)-manifoldM each fiber of which is shape equivalent to N is an approximate fibration. All Hopfian manifolds with Hopfian fundamental group and nonzero Euler characteristic are known to be codimension-2 orientable fibrators. This paper gives a partial answer the following question: is every closed manifold N with π1(N) Hopfian and nonzero Euler characteristic χ(N) ≠ 0 a codimension-2 fibrator? The main result states that, if χ(N) ≠ 0 and π1(N) is finite, then N is a codimension-2 fibrator.