  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.
