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

Abstract:

Melvin has shown that closed 4-manifolds that arise as $S^{2}$-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 $S^{2}\times E^{2}$ or $S^{2}\times \mathbb {H}^{2}$ [depending on whether $\chi (M)=0$ or $\chi (M)<0$]. Conversely a geometric closed, connected 4-manifold $M$ of type $S^{2}\times E^{2}$ or $S^{2}\times \mathbb {H}^{2}$ is the total space of an $S^{2}$-bundle over a closed, connected aspherical surface precisely when its fundamental group $\Pi _{1}(M)$ is torsion free. Furthermore the total spaces of $\mathbb {RP}^{2}$-bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold $M’$ is the total space of an $\mathbb {RP}^{2}$-bundle if and only if $\Pi _{1}(M’)\cong \mathbb {Z}/2\mathbb {Z}\times K$ where $K$ is torsion free.