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Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From Chern-Weil theory it is known that the Chern class of such a line bundle vanishes in rationally. On the other hand, one may construct from each representation $\rho: \pi_1(X) \rightarrow U(1)$ a line bundle $E_{\rho}$ and a mapping which associates $\rho$ with $c_1(E_{\rho})$. This mapping is a homomorphism where the kernel of this homomorphism detects trivial line bundles and the image the Chern classes.

Moreover, it is known from the theorem of Ambrose-Singer that the holonomy group of a flat bundle is discrete.

So my question's are: Given the discrete holonomy group of a flat line bundle, is it possible to compute its Chern class? More generally, is this possible for flat complex vector bundles on Riemann surfaces?

Another question which is related: How much of the topological information can one detect from the holonomy bundle associated to the holonomy group?

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For one of your questions, the answer is simple. The second integral cohomology of Riemann surface is torsion free, so if $c_1$ vanishes rationally, it vanishes. This applies to flat bundles. – Donu Arapura Apr 8 '11 at 12:27
For computing integral first Chern classes of flat line bundles, see my question here :… – Andy Putman Apr 8 '11 at 14:33

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