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Suppose we have a 3-manifold which is triangulated .what can we deduce if we know that the holonomy map is the identity map around each vertex of the triangulation? (Here we imagine the covering space of the manifold as the space of homotopy classes of paths in M which are started from a fixed base point ).

Reference :Geometric structures on low-dimensional manifolds , S.choi , Thoerem 3.

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 This question really is vague. But if I'm not mistaken as to what you could want as an answer, the fact that locally the holonomy is trivial does not mean anything from a global viewpoint, as to me the global holonomy representation could be potentially anything. – Loïc Teyssier Jan 12 at 11:18
Loic is right: If $M$ is an open connected oriented 3-manifold and $\rho: \pi_1(M)\to PGL(3, {\mathbb R})$ is a homomorphism, then there exists a real-projective structure on $M$ with the holonomy $\rho$. On the other hand, understanding possible holonomy homomorphisms of closed real-projective 3-manifolds is a very hard open problem. – Misha Jan 12 at 16:11
What do we have to prove if we want to say that a topologically triangulated manifold M admits a real projective structure such that the topological triangulations is a projective one? – DAVID Jan 12 at 17:45
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 David: The latter is actually a real question, very different from the one you originally asked. Therefore, I suggest you make it a separate question. – Misha Jan 12 at 18:00

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