1999/01/01 by A. B. Goncharov · 1 citation
Mathematics · #Geometric and Algebraic Topology #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algorithm #Mathematics #Artificial intelligence #Computer science
paper · pdf · doi:10.1090/s0894-0347-99-00293-3
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
Two different constructions of an invariant of an odd-dimensional hyperbolic manifold with values in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript 2 n minus 1 Baseline left-parenthesis double-struck upper Q overbar right-parenthesis circled-times double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊗ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K2n-1( \mathbb Q)⊗ \mathbb Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are given. We prove that the volume of the manifold equals the value of the Borel regulator on this invariant. The scissors congruence groups in noneuclidean geometries are studied and related to mixed Tate motives and algebraic K-theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We contribute to the general theory of mixed Hodge structures by introducing for Hodge-Tate structures the big period map with values in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C circled-times double-struck upper C Superscript asterisk Baseline left-parenthesis n minus 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo> ⊗ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb C ⊗ \mathbb C^*(n-2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .