On the motivic description of truncated fundamental group rings

Abstract

A topological theorem that appears in a paper by Deligne-Goncharov (and which they attribute to Beilinson) states the following. Let $(X,*)$ be a path connected pointed space with a reasonable topology and denote by $I$ the augmentation ideal of its fundamental group ring. Then for every field F and positive integer n, the space of F-valued linear forms on $ I/I^{n+1}$ is naturally isomorphic to $H^n(X^n,X(n,*); F)$, where $X(n,*)$ is an explicitly defined subspace of $X^n$. We here construct a simple isomorphism between $I/I^{n+1}$ and $H_n(X^n,X(n,*); \mathbf{Z})$ and express the maps that define the Hopf algebra structure on the $I$-adic completion of the fundamental group ring of $(X,*)$ in these terms.