\(S^2\)-bundles over 2-orbifolds

Jonathan A. Hillman

Abstract

Let \(M\) be a closed 4-manifold with \(\pi=\pi_1(M)\not=1\) 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 \(RP^2\) 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.