Partial solutions of the bad orbifold conjecture (Q1924650)

An \(n\)-orbifold is a topological space locally modelled on (an open set in \(\mathbb{R}^n)\)/(a finite group action) and each point of it is provided with isotropy data. An orbifold is called good if it has a covering orbifold which is a manifold and called bad otherwise. A 2-orbifold is called a turnover if its underlying topological space is a 2-sphere and the singular set consists of three points. \textit{Thurston} [The geometry and topology of 3-manifolds, Mimeographed notes, Princeton Univ., 1978] conjectured that a 3-orbifold is good if it has no bad spheres. In this paper, the author proved this conjecture on a special 3-orbifold which (1) has no turnovers and no bad spheres, and (2) each nontrivial local groups of which is \(\mathbb{Z}_2\) and has no bad spheres. The results are obtained by constructing manifold orbi-covering of the orbifolds.