Suhyoung Choi in his article " Geometric structures on low-dimensional manifolds " proposed a concept that I haven't seen anywhere else. He says that if we have a triangulated topological 3-manifold t and its vertex links (linking spheres of each vertex of the triangulation) then the triangulation induces a triangulation on each link sphere. Also he introduced "vertex diagrams" which I can't understand either. Thurston also says that by having the gluing information of a triangulation, the link of vertexes also has a triangulation. Suppose we have two different triangulated linking sphereS and we want to pair their vertexes. Choi says that a vertex on the first sphere paired with another vertex on the second sphere and the paired vertexes have the same number of edges starting from it. Also every edge ending at a vertex is paired with an edge ending at the paired vertex. The pairing corresponds to an edge of the triangulation of M. I just can't see the correspondence.
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1$\begingroup$ David, this is standard terminology for triangulated manifolds and it occurs to some extend in all the literature. Technically what the author is describing is a PL-compatible triangulation of a PL-manifold. The concept originated with smoothly-compatible triangulations (for smooth manifolds) but it's more natural in the PL setting. $\endgroup$– Ryan BudneyCommented Nov 5, 2012 at 20:10
3 Answers
As far as I can see, what Choi is doing is just the following. Each vertex $i$ of the original triangulation has a sphere as its link. I'm accustomed to thinking of the sphere as consisting of some of the simplices of the triangulation, but for Choi's purposes it seems better to view the link as a much smaller sphere $S_i$ around $i$. Each edge (of your original triangulation) incident to $i$ pierces the link at a point; each triangle incident to $i$ cuts $s$ in an arc joining two of those points; each tetrahedron incident to $i$ covers a triangular patch in $S_i$. Those intersections of $S_i$ with the faces of your triangulation that are incident to $i$ constitute a triangulation of $S_i$.
Now each edge of your original triangulation, say an edge $\{i,j\}$, determines in this way a vertex in the triangulation of $S_i$ and also a vertex in the triangulation of $S_j$. Those two vertices in two link spheres are to be paired with each other. Note that different vertices on $S_i$ are not paired with vertices of the same $S_j$; in fact, the relevant $S_j$ is different for each of these vertices (because the corresponding edges of the original triangulation connect $i$ to different vertices $j$).
Hi. There are also some errata for this paper. See http://mathsci.kaist.ac.kr/~schoi/research.html I am sorry that the paper is so badly written.
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1$\begingroup$ The answer to the initial question is that in 2-dimension projective links, vertex diagrams gives us 2-dim projective structures that have trivial holonomy around each vertex. This is true for all the projective spheres for vertices. Then only possibility is that the holonomy around each edge in the 3-dim case is also trivial. Also the fact that the projective invariants determines the projective structure is needed. We also need to assume generic situations for the link diagrams. $\endgroup$ Commented Feb 19, 2013 at 6:16
"Real projective structures on 3-orbifolds and projective invariants", Melbourne Talk, May 18, 2009. This is a pdf file of my presentation in my homepage. (http://mathsci.kaist.ac.kr/~schoi/mel2.pdf) This has some additional material on this topic.