S2-bundles over 2-orbifolds

Let M be a closed 4-manifold with π = π1(M) 6= 1 and π2(M) ∼= Z, and let u : π → Aut(π2(M)) be the natural action. If π ∼= Ker(u) × Z/2Z then M is homotopy equivalent to the total space of an RP -bundle over an aspherical surface. We show here that if π is not such a product then M is homotopy equivalent to the total space of an S-orbifold bundle over a 2orbifold B. There are at most two such orbifold bundles for each pair (π, u). If B is the orbifold quotient of the orientable surface of genus g by the hyperelliptic involution there are two homotopy types of such orbifold bundles and only one of these is geometric. Every closed 4-manifold with geometry S × E or S × H has a foliation with regular leaves S or RP . The leaf space of such a foliation may be regarded as a compact 2-orbifold. If the regular leaves are S the singularities of this orbifold are cone points of order 2 or reflector curves, and the projection to the leaf space is an orbifold bundle projection, with general fibre S. If there are no exceptional leaves the projection is a bundle projection, and the total space is geometric. (See Theorem 10.9 of [2].) Each pair (π, u) where π = π(B) is a 2-orbifold group and u : π → Z/2Z is an epimorphism with torsion-free kernel is realized by a standard geometric manifold Mst. In §1 we review the key invariants that we shall use, and consider aspects of the cup-product in H(Mst;F2). In §2 we show that if M is any 4-manifold realizing (π, u) then k1(M) = k1(Mst, and that if χ(Mst) is even and B has cone points then v2(Mst) = U , where U ∈ H(π;F2) = Hom(π, Z/2Z) corresponds to the action u. In §3 we consider local models for orbifold bundle projections, and in §4 we show that there are at most two 4-manifolds M which are total spaces of orbifold bundles over B with regular fibre S and action u on π2(M) ∼= Z. The base orbifold B must have a nonsingular double cover. In particular, its singular locus consists of cone points of order 2 and reflector curves. If B has an “untwisted” reflector curve, the 1991 Mathematics Subject Classification. 57N13.