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 π₁(M,x) can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called the relative solvable completions   of π₁(M,x). We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.