## $S^2$-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.

: Primary 57N13.

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 Wednesday, September 22, 2010