$S^2$-bundles over 2-orbifolds

Details $S^2$-bundles over 2-orbifolds $S^2$-bundles over 2-orbifolds

2010-08-25

arxiv.org/abs/1008.4186v3

Mathematics

Geometric Topology

We have completed the determination of which $S^2$-orbifold bundles are geometric, and also computed the second Wu class for e

Scientific paper

Let $M$ be a closed 4-manifold with $\pi_2(M)\cong{Z}$. 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 $\pi=\pi_1(M)\not=1$ there are at most two such bundle spaces with given action $u:\pi\to{Aut}(\pi_2(M))$. The bundle space has the geometry $\mathbb{S}^2\times\mathbb{E}^2$ (if $\chi(M)=0$) or $\mathbb{S}^2\times\mathbb{H}^2$ (if $\chi(M)<0$), except when $B$ is orientable and $\pi$ is generated by involutions, in which case the action is unique and there is one non-geometric orbifold bundle.

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