Finite groups and approximate fibrations

A closed connected n-manifold N is called a codimension-2 fibrator (codimension-2 orientable fibrator, respectively) if every proper map p :M→B on an (orientable, respectively) (n+2)-manifold M each fiber of which is shape equivalent to N is an approximate fibration. The aim of this paper is to prove the following three statements: N is a hopfian manifold with |H1(N)|⩽2, then N is a codimension-2 orientable fibrator. f N is a closed manifold whose fundamental group is isomorphic to a finite product of Z2rʹs for some r, then N is a codimension-2 fibrator. Let N be a hopfian n-manifold with H1(N)≈Z2. If the commutator subgroup [π1(N),π1(N)] of π1(N) is a hyperhopfian group, then N is a codimension-2 fibrator. thod used in (i) and (ii) induces the following: If a codimension-2 PL fibrator N satisfies that both π1(N) and πk−1(N) are finite and that πi(N)=0 for 2⩽i⩽k−2, then N is a codimension-k PL fibrator.