$S^2$-bundles over aspherical surfaces and 4-dimensional geometries

Abstract

Melvin has shown that closed 4-manifolds that arise as S2-bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries S2 ×E2 or S2 ×H2 [depending on whether χ(M) = 0 or χ(M) < 0]. Conversely a geometric closed, connected 4-manifold M of type S2 × E2 or S2 × H2 is the total space of an S2-bundle over a closed, connected aspherical surface precisely when its fundamental group Π1(M) is torsion free. Furthermore the total spaces of RP-bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifoldM ′ is the total space of an RP-bundle if and only if Π1(M ′) ∼= Z/2Z×K where K is torsion free. Introduction Melvin [Me] has shown that closed 4-manifolds that arise as S-bundles over closed aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. Moreover one may construct ξ from w(ξ). See [Me, Structure Lemma]. In particular there are two orientable 4-manifolds that arise as total spaces of S-bundles over each closed surface and are distinguished by whether or not w2(ξ) = 0. Ue [Ue] has shown that the total spaces M of such orientable bundles over the surfaces of Euler characteristic χ = 0 have the structures of Seifert 4-manifolds and are geometric. There are just two S-bundles over S, the product bundle and S2×S2; the latter is not geometric. There are four S-bundles over RP, each of which admits an S2×S2 geometry and is distinguishable by its Stiefel-Whitney classes [H1]. There are four S-bundles over each closed orientable surface B with Euler characteristic χ(B) ≤ 0, six S-bundles over the Klein bottle Kb and eight S-bundles over each closed non-orientable surface B with χ(B) ≤ 0, [Me]. The aim of this paper is to prove the following: Main Theorem. (1) A closed S2×E2or S2×H2-manifold M is the total space of an S-bundle over some aspherical surface precisely when its fundamental group Π1(M) is torsion free. Conversely all total spaces of such bundles are geometric. Received by the editors May 10, 1996. 1991 Mathematics Subject Classification. Primary 57N50; Secondary 57N13, 55R25.