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(Note that I've edited the main body of the question to make it clear for other readers.)

Fix a principal $G$-bundle $\rho: P \rightarrow X$ and fix a point $p \in P_x$ in the fiber above $x \in X$. If $\rho$ is equipped with a flat connection $\omega$, we get a surjective homomorphism $\pi_1(X,x) \rightarrow \text{Hol}_p(\omega)$.

  • When is the map $\pi_1(X, x) \rightarrow \text{Hol}_p(\omega) \hookrightarrow G \rightarrow \text{Inn}(G)$ surjective?

The motivation for this question comes from the fact that such a composition is surjective in the case of ``parallel transport'' of fundamental groups in $X$ along curves changing basepoints.

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This question isn't very clear to me - what exactly do you mean by "force the situation above"? Also, my impression is that the only natural topology on $\pi_1(M)$ is the discrete topology. (I say that since it seems like $\pi_1(M)$ is the component group of the loop group of $M$.) –  Peter Samuelson Sep 28 '11 at 4:03
    
Voting to close until the question actually makes sense. –  Igor Rivin Sep 28 '11 at 9:38
    
I've made the appropriate changes to the question. Does it make sense now? –  Alexander Moll Sep 29 '11 at 2:25
1  
The typical case is that the fundamental group is countable and $Inn(G)$ is uncountable in which case your map is never surjective. –  Johannes Ebert Sep 29 '11 at 7:18
    
Peter, sets of connected components may have a non-trivial totally disconnected topology, e.g. the Cantor set –  Fernando Muro Sep 29 '11 at 10:40
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