Polylogarithms and motivic Galois groups

This paper is an enlarged version of the lecture given at the AMS conference “Motives” in Seattle, July 1991. More details can be found in [G2]. My aim is to formulate a precise conjecture about the structure of the Galois group Gal (MT (F )) of the category MT (F ) of mixed Tate motivic sheaves over Spec F , where F is an arbitrary field. This conjecture implies (and in fact is equivalent to) a construction of complexes Γ(F, n)Q that should satisfy all the Beilinson-Lichtenbaum axioms modulo torsion. In particular, we get a hypothetical description of Kn(F )⊗Q by generators and relations that generalize the definition of Milnor’s K-groups. In the case when F is a number field this together with the Borel theorem implies