Seifert manifolds with fiber spherical space forms

We study the Seifert fiber spaces modeled on the product space $S^3 \times \mathbb {R}^2$. Such spaces are ``fiber bundles'' with singularities. The regular fibers are spherical space-forms of $S^3$, while singular fibers are finite quotients of regular fibers. For each of possible space-form groups $\Delta$ of $S^3$, we obtain a criterion for a group extension $\varPi$ of $\Delta$ to act on $S^3 \times \mathbb {R}^2$ as weakly $S^3$-equivariant maps, which gives rise to a Seifert fiber space modeled on $S^3 \times \mathbb {R}^2$ with weakly $S^3$-equivariant maps $\mathrm {TOP}_{S^3}(S^3 \times \mathbb {R}^2)$ as the universal group. In the course of proving our main results, we also obtain an explicit formula for $H^2(Q; \mathbb {Z})$ for a cocompact crystallographic or Fuchsian group $Q$. Most of our methods for $S^3$ apply to compact Lie groups with discrete center, and we state some of our results in this general context.