Recently, I'm reading the paper "**Analytic structures on the space of flat vector bundles over a compact Riemann surface**" by Gunning. In the introduction of this paper, Gunning says that *the set $H^1(M,PL(n, \mathbb{C}))$ of projectively flat bundles over a compact Riemann surface $M$ has $n$ components, which can be put in one-to-one correspondence with one another.* Then he claims that this property follows directly from the short exact sequence(the details are omitted) 
$$0 \rightarrow \mathbb{Z}_n \rightarrow SL(n, \mathbb{C}) \rightarrow PL(n, \mathbb{C}) \rightarrow 0,$$
where $\mathbb{Z}_n = \mathbb{Z}/(n\mathbb{Z})$.

I think we could obtain the following exact sequence of cohomology groups from the above one
$$0 \rightarrow H^0(M,\mathbb{Z}_n) \rightarrow H^0(M, SL(n, \mathbb{C})) \rightarrow H^0(M,PL(n,\mathbb{C})) \rightarrow H^1(M,\mathbb{Z}_n) \rightarrow H^1(M, SL(n, \mathbb{C})) \rightarrow H^1(M,PL(n,\mathbb{C})) \rightarrow H^2(M,\mathbb{Z}_n)$$
since $\mathbb{Z}_n$ is in the center of $SL(n, \mathbb{C})$. Form the connectness of $M$, we have (am I right?)
$$H^0(M,\mathbb{Z}_n) = \mathbb{Z}_n, H^0(M, SL(n, \mathbb{C}))=SL(n, \mathbb{C}), H^0(M,PL(n,\mathbb{C}))=PL(n,\mathbb{C}), H^2(M,\mathbb{Z}_n)=\mathbb{Z}_n.$$

Therefore, we get an exact sequence
$$0 \rightarrow H^1(M,\mathbb{Z}_n) \rightarrow H^1(M, SL(n, \mathbb{C})) \rightarrow H^1(M,PL(n,\mathbb{C})) \xrightarrow{\delta} \mathbb{Z}_n$$

I guess these $n$ components arise from the preimages of $\delta$. But how to prove 

 1. The map $\delta$ is surjective.
 2. For any $a_1, a_2 \in \mathbb{Z}_n$, there exists a natural one-to-one correspondence between $\delta^{-1}(a_1)$ and $\delta^{-1}(a_2)$.

Furthermore, we know that for any projective bundle $\phi$ over a compact Riemann surface $M$, there always exists a holomorphic vector bundle $E$ such that $\phi = \mathbb{P}(E)$. Hence, if $\phi \in H^1(M,PL(n,\mathbb{C}))$, we can find a holomorphic vector bundle $E$ with $\phi = \mathbb{P}(E)$ over $M$. But we don't know whether $E$ is flat or not (or $\deg(E)$ is zero or not). I think $\delta(\mathbb{P}(E)) = [\deg(E)] \in \mathbb{Z}_n$.  Who can help me prove it or give a counterexample? Thank you very much!