$S^2$-bundles over 2-orbifolds

Abstract

Let $M$ be a closed 4-manifold with $π_2(M)\cong{Z}$. Then $M$ is homotopy equivalent to either $CP^2$, or the total space of an orbifold bundle with general fibre $S^2$ over a 2-orbifold $B$, or the total space of an $RP^2$-bundle over an aspherical surface. If $π=π_1(M)\not=1$ there are at most two such bundle spaces with given action $u:π\to{Aut}(π_2(M))$. The bundle space has the geometry $\mathbb{S}^2\times\mathbb{E}^2$ (if $χ(M)=0$) or $\mathbb{S}^2\times\mathbb{H}^2$ (if $χ(M)<0$), except when $B$ is orientable and $π$ is generated by involutions, in which case the action is unique and there is one non-geometric orbifold bundle.