The Grassmannian complex and Goncharov's motivic complex in weight 4

TL;DR

This paper explores the relationship between the Grassmannian complex and Goncharov's motivic complex in weight 4, providing partial morphisms that connect geometric configurations to motivic cohomology.

Contribution

It constructs three out of four maps linking the Grassmannian complex to Goncharov's weight 4 motivic complex, advancing understanding of higher weight motivic structures.

Findings

Partial morphism in weight 4 constructed

Connections established between geometric configurations and motivic complexes

Progress towards understanding Goncharov's conjectural complexes in higher weights

Abstract

For a field $F$ and a given integer $n>1$, Goncharov has given a complex $\Gamma_F(n)$ which he calls motivic and which he expects to rationally compute the weight $n$ motivic cohomology of $\text{Spec }F$, and hence its algebraic $K$-groups in Adams weight $n$, and he was also led to---conjecturally quasiisomorphic---`thickened' complexes thereof. These complexes involve tensor products of higher Bloch groups, the latter having been linked to the geometry of certain configurations in Goncharov's proof of Zagier's Polylogarithm Conjecture for weight 3, and an analogous picture has long been envisioned by Goncharov for higher weight as well. We provide a partial morphism in weight 4 by giving three out of four maps for configurations in general position.