differential topology - Understanding iterated integrals

As mentioned by Emerton, iterated integrals only work well for unipotent representations of $pi_1(X,x)$.

The reason for this is that differential forms are abelian objects: for paths $gamma_i$, and a closed 1-form
$alpha in Omega^1(X)$,

$$int_{gamma_1gamma_2} alpha = int_{gamma_1} alpha + int_{gamma_2} alpha = int_{gamma_2gamma_1} alpha$$
That and homotopy invariance implies that integration induces a pairing
$$int: H^1(Omega(X)) otimes mathbb{Q} [pi_1(X,x)]^{ab} to mathbb{C}$$
where $mathbb{Q}[pi_1(X,x)]^{ab} = H_1(X;mathbb{Q})$.

By considering iterated integrals, we can go one step further. The above pairing has a generalization as

$$int: H^0(Ch^{leq n}(X)) otimes mathbb{Q}[pi_1(X,x)]/J_x^{n+1} to mathbb{C}$$
where $Ch^{leq n}(X)$ is the lenght $leq n$ part of Chen's complex and $J$ is the augmention ideal generated by the $(gamma-1)$.
So iterated integrals describe the pro-unipotent (Malcev) completion $pi_1^{uni}(X,x)$ of $pi_1(X,x)$.
And $varinjlim_n H^0(Ch^{leq n}(X))$ can be thought of as the Hopf algebra of functions on the (pro-unipotent) de Rham fundamental group. One can also define a Hodge and weight filtration and get a pro-mixed Hodge structure on $varprojlim mathbb{Q}[pi_1(X,x)]/J^n$.

This allows to extend the correspondance between unipotent local systems and unipotent representations the fundamental group to the de Rham and even the Hodge or motivic setting. Of course there are technical conditions for things to go smoothly. Basically one needs $X$ to be a unipotent $K(pi,1)$ in the sense that $H^i (pi_1^{uni}(X,x),mathbb{Q}) to H^i(X,mathbb{Q})$ is an isomorphism. In the language of rational homotopy theory this corresponds to 1-minimality.

PS: It is not clear how to proceed to go beyond the unipotent setting. I think Hain, Matsumoto and Terasoma have a generalization of the bar construction that works for more general "relative completions" but nothing has been published yet.

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