Iterated Integrals and higher order invariants

TL;DR

This paper explores the relationship between free iterated integrals and higher order invariants, demonstrating that certain modules are generated by closed iterated integrals, building on Chen's foundational work.

Contribution

It establishes that the restriction morphism from free to loop iterated integrals is onto and that higher order invariants are generated by free closed iterated integrals.

Findings

Restriction morphism from free to loop integrals is onto

Higher order invariants are generated by free closed iterated integrals

Builds on Chen's work to connect iterated integrals and invariants

Abstract

We deduce from the work of Chen, that the restriction morphism from closed free iterated integrals to closed iterated integrals on loops is onto. We use this to show that the module of higher order invariants of smooth functions is generated by free closed iterated integrals.