{"id":{"repo_id":"durham","oai_identifier":"oai:etheses.durham.ac.uk:586"},"canonical_url":"https://search.dev.ndltd.org/etd/durham/oai:etheses.durham.ac.uk:586","repository":{"repo_id":"durham","name":"Durham University","base_url":"http://etheses.dur.ac.uk/cgi/oai2"},"display":{"title":"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 $F$-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.","abstract_html":"In this thesis we consider the Grassmannian complex of projective configurations in weight 2 and 3, and Cathelineau&#x27;s infinitesimal polylogarithmic complexes as well as a tangential complex to the famous Bloch-Suslin complex (in weight 2) and to Goncharov&#x27;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 <span class=\"etd-inline-math\">&beta;<sup>D</sup><sub>2</sub>(F)</span>, which is an intermediate structure between a $\\varmathbb{Z}$-module <span class=\"etd-inline-math\">\\mathcal{B}<sub>2</sub>(F)</span> (scissors congruence group for $F$) and Cathelineau&#x27;s $F$-vector space <span class=\"etd-inline-math\">&beta;<sub>2</sub>(F)</span> which is an infinitesimal version of it. The structure of <span class=\"etd-inline-math\">&beta;<sup>D</sup><sub>2</sub>(F)</span> is also infinitesimal but it has the advantage of satisfying similar functional equations as the group <span class=\"etd-inline-math\">\\mathcal{B}<sub>2</sub>(F)</span>. We put this in a complex to form a variant of Cathelineau&#x27;s infinitesimal complex for weight 2. Furthermore, we define <span class=\"etd-inline-math\">&beta;<sub>3</sub><sup>D</sup>(F)</span> 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&#x27;s triple-ratios as the product of two projected cross-ratios. Furthermore, we extend Siegel&#x27;s cross-ratio identity ([21]) for $2\\times2$ determinants over the truncated polynomial ring <span class=\"etd-inline-math\">F[\\varepsilon]<sub>\\</sub>nu:=F[\\varepsilon]/\\varepsilon<sup>\\</sup>nu</span>. We compute cross-ratios and Goncharov&#x27;s triple-ratios in <span class=\"etd-inline-math\">F[\\varepsilon]<sub>2</sub></span> and <span class=\"etd-inline-math\">F[\\varepsilon]<sub>3</sub></span> and use them extensively in our computations for the tangential complexes. We also verify a &#x27;&#x27;projected five-term&#x27;&#x27; relation in the group <span class=\"etd-inline-math\">T\\mathcal{B}<sub>2</sub>(F)</span> which is crucial to prove one of our central statements Theorem 4.3.3.","abstract_has_math":true,"creators":["Siddiqui, Raziuddin"],"institution":"Durham University","degree_name":"PhD","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-24T02:11:12Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Siddiqui, Raziuddin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010"]},{"key":"dc:date.issued","label":"Date","values":["2010"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematical Sciences, Department of"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Durham University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://etheses.durham.ac.uk/id/eprint/586/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://etheses.durham.ac.uk/id/eprint/586/1/final_soft_thesis.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["Configuration Complexes and Tangential and Infinitesimal versions of Polylogarithmic Complexes"]}]}],"canonical_facts":{"dc:creator":["Siddiqui, Raziuddin"],"dc:date":["2010"],"dc:date.issued":["2010"],"dc:description.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 $F$-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."],"dc:format":["text"],"dc:identifier.uri":["https://etheses.durham.ac.uk/id/eprint/586/1/final_soft_thesis.pdf"],"dc:publisher.department":["Mathematical Sciences, Department of"],"dc:publisher.institution":["Durham University"],"dc:relation.isreferencedby":["https://etheses.durham.ac.uk/id/eprint/586/"],"dc:title":["Configuration Complexes and Tangential and Infinitesimal versions of Polylogarithmic Complexes"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-24T02:11:12Z"}