Loops, Holonomy and Signature

Abstract

We show that there is a topology on certain groups of loops in Euclidean space such that these groups are embedded in a Fr\'echet-Lie group which is the structural group of a principal bundle with connection whose holonomy coincides with the Chen signature map. We also give an alternative geometric new proof of the Chen signature theorem and a generalization of this theorem in classes strictly containing the one originally considered by Chen.