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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".

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Your question is followed by a quote which seems to give the answer your question ... – Yemon Choi Dec 22 '12 at 8:05
up vote 6 down vote accepted

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.

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I am not sure that the Orbifold has to be good. It seems that it only has to be "locally good" in which case, we could take the local universal cover and apply the same recipe as in the answer. – Spice the Bird Dec 22 '12 at 13:11
why is the link of the vertex v the formula you said? – DAVID Dec 22 '12 at 13:36
Spice the Bird - this is certainly true. But the good case is particularly easy to visualise. – HJRW Dec 23 '12 at 17:36
David - well, your question contains the definition. All my answer does is attempt to explicate it. – HJRW Dec 23 '12 at 17:37
so I think I haven't understand the link of a vertex of a manifold .what is your definition of a link of a vertex of a manifold? – DAVID Dec 27 '12 at 4:02

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