Iterated integrals, fundamental groups and covering spaces

Abstract:

Differential $1$-forms are integrated iteratedly along paths in a differentiable manifold $X$. The purpose of this article is to consider those iterated integrals whose value along each path depends only on the homotopy class of the path. The totality of such integrals is shown to be dual, in an appropriate sense, to the “maximal” residually torsion free nilpotent quotient of the fundamental group ${\pi _1}(X)$. Taken as functions on the universal covering space $\tilde X$, these integrals separate points of $\tilde X$ if and only if ${\pi _1}(X)$ is residually torsion free nilpotent.