Iterated integrals and cohomology

TL;DR

This paper introduces an extended cohomology framework linked to iterated integrals, noncommutative modular symbols, and false theta functions, providing new insights into their algebraic and modular properties.

Contribution

It develops a novel cohomological approach to interpret iterated integrals and relations among multiple zeta values in a modular context.

Findings

Cohomological interpretation of iterated integrals related to modular symbols

Extension of cohomology to include non-classical cocycles

Modular analogues of relations among multiple zeta values

Abstract

We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of false theta functions. We use this construction to obtain a cohomological interpretation of important iterated integrals that arise in that study. In another direction, our approach gives modular counterparts to the long-studied relations among multiple zeta values.