Iterated Integrals in Quantitative Topology

Let X be a simply connected Riemannian manifold. Until now, quantitative topology has used Sullivan's rational homotopy theory as the bridge between geometric information on X and torsion-free homotopy theoretic information on X. In this paper we introduce Chen's iterated integrals on the based loop space of X as a new bridge between these two areas. We give two applications: finding upper bounds for Gromov's distortion of higher homotopy groups on X and also proving the non-existence of homologically non-trivial small-volume cycles in the space of loops on X of length at most L.