2-Sphere Bundles Over Compact Surfaces

Abstract

Closed 4-manifolds which fiber over a compact surface with fiber a sphere are classified, and the fibration is shown to be unique (up to diffeomorphism). It is well known that there are at most two orientable 4-manifolds which fiber over a given compact surface with fiber the 2-sphere S2. (There is exactly one if the surface has nonempty boundary, and two if it is closed.) If the orientability condition is dropped, then the situation becomes more involved. In particular the (mod 2) intersection pairing is no longer sufficient to distinguish among the mani- folds that arise. One must also consider the ?Tl-action on g2 and the peripheral structure. The purpose of this note is to classify all 4-manifolds (orientable or not) which are total spaces of S2-bundles over compact surfaces. We shall work in the smooth category. Since Diff(S2) deformation retracts to 0(3), we may assume that all bundles that arise have 0(3) as structure group. Along the way it is shown that the bundle structures are unique. That is, if any two 4-manifolds, fibered as above, are diffeomorphic, then there is a fiber preserving diffeomorphism between them which is orthogonal on fibers. Our interest in S2-bundles arose in the study of Lie group actions (in particular of S0(3)) on 4-manifolds. The results obtained here are used in the equivariant classification of such actions (MP).