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Let $X$ be a connected compact Riemann surface. How does one go on proving that the set of PGL($n,\mathbb{C}$)-bundles over $X$ is topologically classified by $\pi_1(PGL(n,\mathbb{C}))$? Is it true for more general groups than PGL($n,\mathbb{C}$)? Thanks in advance.

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    $\begingroup$ Did you mean $\mathrm{Hom}(\pi_1(X), \mathrm{PGL}_n(\mathbb C))$ instead of $\pi_1(\mathrm{PGL}_n(\mathbb C))$? The latter does not depend on $X$, which seems weird. $\endgroup$ – Arun Debray Apr 20 '18 at 19:55
  • $\begingroup$ @ArunDebray : No, I meant $\pi_1(PGL(n,\mathbb{C})$ only. $\endgroup$ – yojusmath Apr 21 '18 at 6:40
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    $\begingroup$ Classification in terms of $\pi_1(PGL(n,\mathbb{C}))$ works for $X=\mathbb{P}^1$. For genus $\geq 1$, classification is in terms of ${\rm Hom}(\pi_1(X),{\rm PGL}_n(\mathbb{C}))$ as in the comment of @ArunDebray. Arguments for that can be extracted from the answer to this MO-question: mathoverflow.net/questions/20764 $\endgroup$ – Matthias Wendt Apr 21 '18 at 10:26

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