$S^2$-bundles over 2-orbifolds

Jonathan A. Hillman

Abstract

Let $M$ be a closed 4-manifold with $\pi ={\pi }_1(M)\ne 1$ and ${\pi }_2(M)\cong Z$, and let $u:\pi \to \mathrm{Aut}({\pi }_2(M))$ be the natural action. If $\pi \cong \mathrm{Ker}(u)\times Z/2Z$ then $M$ is homotopy equivalent to the total space of an $R{P}^2$ bundle over an aspherical surface. We show that if $\pi$ is not such a product then $M$ is homotopy equivalent to the total space of an $S^2$-orbifold bundle over a 2-orbifold $B$. There are at most two such orbifold bundles for each pair $(\pi ,u)$. If $B$ is the orbifold quotient of an orientable surface by the hyperelliptic involution there are two homotopy types of such bundles and only one of these is geometric.

Keywords: geometry. 4-manifold. orbifold. $S^2$-bundle.

AMS Subject Classification: Primary 57N13.