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Durham e-Theses
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Configuration Complexes and Tangential and Infinitesimal versions of Polylogarithmic Complexes

SIDDIQUI, RAZIUDDIN (2010) Configuration Complexes and Tangential and Infinitesimal versions of Polylogarithmic Complexes. Doctoral thesis, Durham University.

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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 $F$-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.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2010
Copyright:Copyright of this thesis is held by the author
Deposited On:24 Jan 2011 14:55

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