Question

I have encountered iterated integrals on papers dealing with multizeta values, polylogarithms etc.. Since then I am trying to figure out the motivations and purpose of the theory.

It seems the defintions and methods go back to K.-T. Chen. The integrals seem to converge like an exponential series. He published many papers on this topic. Some of these(as seen in his collected works) seem to relate to path spaces, loops spaces etc., and their homology/cohomology. Many notions in algebraic topology seem to be carried out in this context using the analytic tool of iterated integral. He calls it a "de Rham theoretical approach" to the fundamental group, etc.. Is this a "de Rham homotopy theory"? Are we able to capture topological properties by repeated integration? In particular I have in mind the "iterated path integrals" paper of K.-T. Chen in mind. There are lots of others too, and some of them are in the Annals; so one cannot question the mathematical importance of the topic.

I am sorry for asking a vague question. I am a beginner on a topic struggling to understand the concepts and motivations behind them. I will be grateful for any pointers towards more understanding, so that I can get started.

Answer 1

The theory of iterated integral gives a mixed Hodge structure on rational homotopy of a variety. In the case of the fundamental group, as far as I know, one can only detect the nilpotent completion of the fundamental group. (At least, this is the only part of $\pi_1$ that people work with in a motivic context --- see e.g. Deligne's paper on the thrice punctured sphere, many papers of Dick Hain, and perhaps Sullivan's original paper from 1977 on rational homotopy theory.)

See recent work of Minhyong Kim for application of these ideas to studying rational points on curves over number fields.

Answer 2

One topological significance of these iterated integrals is that they can be used to model the bar complex on the de Rham cochains of a manifold, and thus in some sense the "de Rham complex" of the manifold's loop space since $H_*(Bar(\Lambda^* X)) \cong H^*(\Omega X)$ [Here I am using $\Lambda$ for the de Rham complex and $\Omega X$ to denote loops]. For a simply-connected space, this bar complex gives a reasonable way to encode the rational homotopy type, though Sullivan models have been more popular. So in particular, one can understand a complete set of homotopy functionals (that is, the linear dual of homotopy groups) using iterated integrals: Milnor and Moore showed that $\pi_*(X) \otimes {\mathbb Q}$ is isomorphic to the Lie algebra of indecomposibles of $H_*(\Omega X; {\mathbb Q})$; thus the linear dual of homotopy is the "Lie coalgebra of coindecomposibles" of the rational loopspace cohomology of $X$ (again given by Chen integrals if you wish). Chen showed early on that these integrals give information about $\pi_1$ as well (through its nilpotent completion).

My coauthor Ben Walter and I have developed a model for rational homotopy types which is close to both the Chen and Quillen models (and compatible with the Sullivan model as well), and clarified the story of functionals on homotopy groups as well. Because of Chen's work, we know that we will have plenty of information to mine in the non-simply connected setting.

Answer 3

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 $\alpha \in \Omega^1(X)$, $$ \int_{\gamma_1\gamma_2} \alpha = \int_{\gamma_1} \alpha + \int_{\gamma_2} \alpha = \int_{\gamma_2\gamma_1} \alpha $$ 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.