TITLE: What does "higher monodromy" tell us about a principal bundle
QUESTION [6 upvotes]: Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to construct such a presentation out of a principal connection on $P$ by taking an appropriate quotient of the holonomy representation but I failed in the past weeks to find a source that defines a purely topological construction of this representation.

Question 1: Where can I find the definition and some elementary topological treatment of the
  monodromy representation associated to a principal bundle?

Extending the classifying map $f$ to a fiber sequence gives:
$$\cdots\to \Omega G \to \Omega E_f \to \Omega X\to G \to E_f \to X \to BG$$
Where $E_f=\{(x,\gamma)\in X \times Y^I:\gamma(1)=f(x)\}$. 

Question 2: How is the map $\Omega X \to G$ above related to the usual notion of monodromy represnetation (which i'm unfamiliar with) of the
  the principal bundle $P\to X?$ How does it relate to the holonomy representation of a given connection?

My naive attempt continues by taking the fundamental group of the sequence above to get the following long exact sequence of groups:
$$\cdots \to \pi_2(G) \to \pi_2(E_f) \to \pi_2(X)\to \pi_1(G) \to \pi_1(E_f)  \to \pi_1(X) \to \pi_0(G)$$
Since the procedure was very natural I imagine the above sequence must yield some important information about the principal bundle at hand. For example ff $X$ is weakly contractible the above gives another motivation for why every fiber bundle on $X$ is trivial.

Question 3: To what extent does the above "higher monodromy sequence" determine the principal bundle $P \to X$?

REPLY [3 votes]: Question 3: To what extent does the above "higher monodromy sequence" determine the principal bundle $P \to X$? 

In general, not at all. There is no loss of generality in replacing $BG$ with an arbitrary connected space $Y$ (every such space is some $BG$, morally for $G = \Omega Y$), so your question is to what extent the homotopy class of a (pointed) map $f : X \to Y$ is determined by the long exact sequence in homotopy for the fiber sequence
$$F \to X \to Y$$
where $F$ is the homotopy fiber. What can happen in general is that $X$ and $Y$ have homotopy groups in degrees that are far apart from each other, so that this sequence tells you nothing about the homotopy class of $f$. 
For example, if $X = BG$ for some discrete group $G$ and $Y = B^3 A$ for some discrete abelian group $A$, then all you learn is that $F$ is a space with $\pi_1(F) \cong G, \pi_2(F) \cong A$, and other homotopy groups trivial, and all of the maps in the long exact sequence are either isomorphisms or zero, regardless of $f$. But of course the group cohomology $H^3(BG, A)$ is interesting in general.