- CiteKey: “BGSV90” - Type: bookSection - Title: “Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane,” - Author: “Beilinson, A. A.; Goncharov, A. B.; Schechtman, V. V.; Varchenko, A. N.;” - Publisher: “Birkhäuser,” - Year: 2007 - Collections: “Syllabus; Talbot 2022,” - Keywords: “Admissible Pair”; “Canonical Isomorphism”; “Free Abelian Group”; “Hodge Structure”; “Hopf Algebra”
Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane
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- URL: https://doi.org/10.1007/978-0-8176-4574-8_6
- URI: http://zotero.org/users/1049732/items/2AIFCMH9
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Abstract
It is known that a group of linear combinations of polytopes in R3 considered up to movements with respect to cutting of polytopes may be embedded into ℝ ⊗ ℝ/2πℤ ⊕ ℝ; this embedding assigns to a polytope its Dehn invariant and volume [C]. The study of motivic cohomology of a projective plane with two distinguished families of projective lines leads to an analogous problem: to describe a group of linear combinations of pairs of triangles on a plane considered up to the action of PGL(3), with respect to a cutting of any triangle of a pair. It turns out that this group is isomorphic up to 12—torsion to B2 ⊕ S2B1, where S2B1 is the symmetric square of the multiplicative group of a ground field, and B2 — the Bloch group of this field. This is the first main result of the paper (see Theorems 2.12, 3.8 and 3.6.2).