The study of 4-manifolds which fibre over the circle benefits from both 4-dimensional surgery and from 3-manifold theory, which may be applied to the fibre and the characteristic map. Unfortunately, there is as yet no satisfactory fibration theorem in this dimension. We shall give a homotopy approximation to such a theorem. By a result of Quinn, an infinite cyclic covering space of a closed n-manifold M is homotopy equivalent to a PDn-1-complex if and only if it is finitely dominated. The fundamental group v of the covering space must then be finitely generated, and χ(M) = 0. It is conceivable that these conditions may suffice when n = 4; we shall see that this is so if also either v is free or v does not contain the centre of π1(M).

In the final section we give conditions for a 4-manifold to be homotopy equivalent to the total space of an S1-bundle over a PD3-complex, and show that these conditions are sufficient if the fundamental group of the PD3-complex is torsion free but not free.

A general criterion

Let E be a connected cell complex and let f : E → S1 be a map which induces an epimorphism f* from π1(E) to Z = π1(S1) with kernel v.