Exponential Iterated Integrals and the Relative Solvable Completion of the Fundamental Group of a Manifold

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen’s iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called relative solvable completions, of the fundamental group. We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.