Configuration Complexes and Tangential and Infinitesimal versions of Polylogarithmic Complexes

Abstract

In this thesis we consider the Grassmannian complex of projective configurations in weight 2 and 3, and Cathelineau's infinitesimal polylogarithmic complexes as well as a tangential complex to the famous Bloch-Suslin complex (in weight 2) and to Goncharov's ``motivic`` complex (in weight 3), respectively, as proposed by Cathelineau [5].

Our main result is a morphism of complexes between the Grassmannian complexes and the associated infinitesimal polylogarithmic complexes as well as the tangential complexes.
In order to establish this connection we introduce an -vector space , which is an intermediate structure between a -module (scissors congruence group for ) and Cathelineau's -vector space which is an infinitesimal version of it. The structure of is also infinitesimal but it has the advantage of satisfying similar functional equations as the group . We put this in a complex to form a variant of Cathelineau's infinitesimal complex for weight 2. Furthermore, we define for the corresponding infinitesimal complex in weight 3. One of the important ingredients of the proof of our main results is the rewriting of Goncharov's triple-ratios as the product of two projected cross-ratios. Furthermore, we extend Siegel's cross-ratio identity ([21]) for determinants over the truncated polynomial ring . We compute cross-ratios and Goncharov's triple-ratios in and and use them extensively in our computations for the tangential complexes. We also verify a ''projected five-term'' relation in the group which is crucial to prove one of our central statements Theorem 4.3.3.