Hutchinson, Kevin; Vlasenko, Masha Lines crossing a tetrahedron and the Bloch group. (English) Zbl 1262.19003 Pragacz, Piotr (ed.), Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4–10, 2010. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-114-9/hbk). EMS Series of Congress Reports, 297-303 (2012). Let \(k\) be an infinite field. For \(n\geq 0\) the group \(G=\mathrm{GL}_2(k)\) acts on the free abelian group \(C(2,n)\) generated by \((n+1)\)-tuples of vectors in \(k^2\) that are pairwise linearly independent. One can form a complex \(\cdots\to C(2,2)\to C(2,1)\to C(2,0)\to0\) of \(G\)-modules and study its hyperhomology spectral sequence. By taking \(G\)-orbits of the \((n+1)\)-tuples for \(n\geq2\) one gets another complex and the ‘lines crossing a tetrahedron’ of the title become visible. As in [A. A. Suslin, Proc. Steklov Inst. Math. 183, 217–239 (1991); translation from Tr. Mat. Inst. Steklova 183, 180–199 (1990; Zbl 0741.19005)] there are connections with Milnor K-theory and the Bloch group. The main theorem describes this in terms of cross ratios.For the entire collection see [Zbl 1245.14003]. Reviewer: Wilberd van der Kallen (Utrecht) MSC: 19D45 Higher symbols, Milnor \(K\)-theory 18G40 Spectral sequences, hypercohomology 19C20 Symbols, presentations and stability of \(K_2\) 19D55 \(K\)-theory and homology; cyclic homology and cohomology Keywords:Bloch group; complexes of configurations; Milnor K-group; cross ratio Citations:Zbl 0741.19005 PDFBibTeX XMLCite \textit{K. Hutchinson} and \textit{M. Vlasenko}, in: Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4--10, 2010. Zürich: European Mathematical Society (EMS). 297--303 (2012; Zbl 1262.19003) Full Text: DOI arXiv