Question

I know the notion of the link of a vertex of a 3-manifold. In his article Geometric structures on low-dimensional manifolds, Suhyoung Choi first defined the notion of "projective triangulation of an orbifold with a projective structure" which is a cellular decomposition of the underlying space induced by a triangulation of its universal cover equivariant with respect to the deck transformation group. How can we generalize the notion "link of a vertex" to the orbifold case? Choi said: in the orbifold case, the link of vertex is a 2-orbifold of positive Euler characteristic. He called this link the "link orbifold".

Answer 1

Let $O$ be the 3-orbifold, $\widetilde{O}$ its universal cover (it sounds like we're assuming that $O$ is 'good', ie has a manifold universal cover) and $\pi_1O$ its fundamental group.

Let $v$ be a vertex covered by a vertex $\tilde{v}$ in the universal cover. Then the link of $v$ is just the 2-orbifold $\mathrm{Link}(\tilde{v})/\mathrm{Stab}_{\pi_1O}(\tilde{v})$.

Because orbifold Euler characteristic is multiplicative and the link is covered by the 2-sphere $\mathrm{Link}(\tilde{v})$, the link does indeed have positive Euler characteristic as claimed.