$S^2$-bundles over 2-orbifolds

TL;DR

This paper classifies certain 4-manifolds with specific homotopy properties as orbifold bundles over 2-orbifolds, detailing their geometric structures and uniqueness conditions based on fundamental group actions.

Contribution

It provides a classification of closed 4-manifolds with $ au_2(M) eq 0$ as orbifold bundles over 2-orbifolds, including geometric and uniqueness results based on fundamental group actions.

Findings

Classifies 4-manifolds with $ au_2(M) eq 0$ as orbifold bundles over 2-orbifolds.

Determines geometric structures: $ ext{S}^2 imes ext{E}^2$ or $ ext{S}^2 imes ext{H}^2$.

Establishes conditions for uniqueness of the bundle structure.

Abstract

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.