S-2-BUNDLES OVER ASPHERICAL SURFACES AND 4-DIMENSIONAL GEOMETRIES

Melvin has shown that closed 4-manifolds that arise as S-2-bundles ove r closed, connected aspherical surfaces are classified up to diffeomor phism by the Stiefel-Whitney classes of the associated bundles. We sho w that each such 4-manifold admits one of the geometries S-2 x E-2 or S-2 x H-2 [depending on whether chi(M) = 0 or chi(M) < 0]. Conversely a geometric closed, connected 4-manifold M of type S-2 x E-2 or S-2 x H-2 is the total space of an S-2-bundle over a closed, connected asphe rical surface precisely when its fundamental group Pi(1)(M) is torsion free. Furthermore the total spaces of RP2-bundles over closed, connec ted aspherical surfaces are all geometric. Conversely a geometric clos ed, connected 4-manifold M' is the total space of an RP2-bundle if and only if Pi(1)(M') congruent to Z/2Z x K where K is torsion free.