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

For a field F F and a given integer n > 1 n>1 , Goncharov (1995) has given a motivic complex Γ F ( n ) \Gamma _F(n) , generalising the Bloch-Suslin complex (Bloch, 2000; Suslin, 1990) for n = 2 n=2 , which he expects to rationally compute the weight n n motivic cohomology of Spec ⁡ F \operatorname {Spec}F , hence its algebraic K K -groups in Adams weight n n , and he was also led to—conjecturally quasiisomorphic—‘thickened’ complexes thereof. These complexes involve tensor products of higher polylogarithm 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 (1995) for higher weight. We provide a partial morphism in weight 4 by giving three out of four maps for configurations in general position. We moreover give partial results for the leftmost square.