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Under what general assumptions $(G,X)$-geometric structures on a manifold $M$ are classified by their holonomies, yielding an injection $$\Psi: \{(G,X)\text{-structures on M}\} \to H(\pi_1(M),G)/G ?$$ When is the image a connected component of $H(\pi_1(M),G)/G$ in the Euclidean (other?) topology? (Probably $(G,X)$-structures need to be "marked".)

I believe that is the case for hyperbolic, Euclidean, affine, projective structures on surfaces.

Is it true in general? If not, what goes wrong? Perhaps extra conditions are necessary, say $RP^2$-structures on surfaces need to be convex.

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    $\begingroup$ This (injectivity) is false for affine and projective (both real and complex) structures. The only cases when I know it works are locally homogeneous Riemannian metrics. $\endgroup$
    – Misha
    Commented Mar 20, 2017 at 13:14
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    $\begingroup$ The map $\Psi$ is a local homeomorphism (if you factor the left side by diffeomorphisms of $M$); see Goldman's beautiful arxiv.org/abs/1003.2759. But there are examples of $(G,X)$ where $\Psi$ is neither injective nor surjective. $\endgroup$
    – Ben McKay
    Commented Mar 20, 2017 at 13:23
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    $\begingroup$ @BenMcKay: Actually, local homeomorphism is yet another misconception, there are examples when it fails (Moebius structures on some Seifert manifolds). See my 1990 survey. $\endgroup$
    – Misha
    Commented Mar 20, 2017 at 17:34
  • $\begingroup$ @Misha: thanks; that is a surprise. $\endgroup$
    – Ben McKay
    Commented Mar 20, 2017 at 18:55
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    $\begingroup$ @BenMcKay: The map is open regardless of the situation but in order to make it locally injective one needs to consider "pointed" geometric structures and $Hom(\pi_1, G)$ instead of the character variety. Bill Goldman promised to have a clean discussion of this somewhere, one day. $\endgroup$
    – Misha
    Commented Mar 20, 2017 at 19:42

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This question was pretty much answered in the comments, so I will just give some relevant references:

  1. Geometric structures on manifolds and varieties of representations, by W. Goldman, Geometry of group representations (Boulder, CO, 1987), 169–198, Contemp. Math., 74, Amer. Math. Soc., Providence, RI, 1988.
  2. Locally homogeneous geometric manifolds, by W. Goldman, Proceedings of the 2010 International Congress of Mathematicians, Hyderabad, India (2010), 717-744, Hindustan Book Agency, New Delhi, India

The above two references describe the general picture pretty well: the moduli space of marked $(G,X)$-structures on a smooth compact manifold is locally modeled on the character variety (in some sense).

In practice, regardless of the properties of the holonomy map, one often uses the structure of the character variety to determine the structure of the moduli space of $(G,X)$-structures. This is exemplified well in Section 4 of this:

Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, by W. Goldman, Handbook of Teichmüller theory. Vol. II, 611-684, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009

W. Goldman has also made available his working book on the subject: Geometric structures on manifolds, which contains many detailed examples.

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