Location of Repository

## 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].\ud \ud Our main result is a morphism of complexes between the Grassmannian complexes and the associated infinitesimal polylogarithmic complexes as well as the tangential complexes.\ud In order to establish this connection we introduce an $F$-vector space $\beta^D_2(F)$, which is an intermediate structure between a $\varmathbb{Z}$-module $\mathcal{B}_2(F)$ (scissors congruence group for $F$) and Cathelineau's $F$-vector space $\beta_2(F)$ which is an infinitesimal version of it. The structure of $\beta^D_2(F)$ is also infinitesimal but it has the advantage of satisfying similar functional equations as the group $\mathcal{B}_2(F)$. We put this in a complex to form a variant of Cathelineau's infinitesimal complex for weight 2. Furthermore, we define $\beta_3^D(F)$ 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 $2\times2$ determinants over the truncated polynomial ring $F[\varepsilon]_\nu:=F[\varepsilon]/\varepsilon^\nu$. We compute cross-ratios and Goncharov's triple-ratios in $F[\varepsilon]_2$ and $F[\varepsilon]_3$ and use them extensively in our computations for the tangential complexes. We also verify a ''projected five-term'' relation in the group $T\mathcal{B}_2(F)$ which is crucial to prove one of our central statements Theorem 4.3.3

Year: 2010
OAI identifier: oai:etheses.dur.ac.uk:586
Provided by: Durham e-Theses

### Citations

1. (2010). Additive polylogarithms and their functional equations,
2. Algebraic cycles and additive dilogarithm,
3. (2003). An additive version of higher chow groups,
4. An extended version of Additive K-theory,
5. (1921). Approximation algebraischer Zahlen,
6. (1970). Commutative Algebra,
7. (1986). Commutative Ring Theory,
8. (1990). Dedekind zeta functions and the algebraic K-theory of ﬁelds,
9. (1996). Deninger’s conjecture on L-functions of elliptic curves at s =
10. Determinantal varieties over truncated polynomial rings,
11. (2004). Euclidean Scissors congruence groups and mixed Tate motives over dual numbers,
12. Explicit construction of characteristic classes,
13. Geometry of Conﬁgurations, Polylogarithms and Motivic Cohomology,
14. (1985). Homology of GLn, characteristic classes and Milnor’s K-theory,
15. (1997). Inﬁnitesimal Polylogarithms, multiplicative Presentations of K¨ ahler Dierentials and Goncharov complexes, talk at the workshop on polylogarthms,
16. (1991). K3 of a ﬁeld and the Bloch group,
17. (2001). Motivic Complexes of Weight Three and Pairs of Simplices
18. On cliques of exceptional units and Lenstra’s constructionof Euclidean ﬁelds,
19. (2002). On Poly(ana)logs I,
20. On the additive dilogarithm,
21. (1991). Polylogarithms and Motivic Galois Groups, Proceedings of the Seattle conf. on motives,
22. (2004). Projective Conﬁgurations, Homology of Orthogonal Groups,
23. Regulators on additive higher chow groups,
24. Remarques sur les Di´ erentielles des Polylogarithmes Uniformes,
25. (2003). The additive dilogarithm, Kazuya Kato’s ﬁftieth birthday,
26. (2007). The tangent complex to the Bloch-Suslin complex,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.