If B, E and F are connected finite complexes and p : E → B is a Hurewicz fibration with fibre homotopy equivalent to F then χ(E) = χ(B)χ(F) and there is an exact sequence π2(B) → π1(F) → π1(E) → π1(B) → 1 in which the image of π2(B) is in the centre of π1(F). These conditions are clearly homotopy invariant.

In this chapter we shall show that a closed 4-manifold M is homotopy equivalent to the total space of a fibre bundle with base and fibre closed surfaces if and only if these conditions on the Euler characteristic and fundamental group hold. When the base is S2 or RP2 we need also conditions on the characteristic classes of M, and in the latter case our results are incomplete when M is nonorientable.

Bundles with base an aspherical surface

If X is a finite complex let E(X) be the monoid of all self homotopy equivalences of X, with the compact-open topology, and if X has been given a base point let E0(X) be the subspace (and submonoid) of base point preserving self homotopy equivalences. The evaluation map from E(X) to X is a Hurewicz fibration with fibre E0(X) [Go68].