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16349 results in Geometry and topology

Page 95 of 818

  • The strong Suslin reciprocity law

  • Part of
  • Daniil Rudenko
  • Journal:
    Compositio Mathematica / Volume 157 / Issue 4 / April 2021
    Published online by Cambridge University Press:
    01 April 2021, pp. 649-676
    Print publication:
    April 2021

  • We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

  • COM volume 157 issue 4 Cover and Back matter

  • Journal:
    Compositio Mathematica / Volume 157 / Issue 4 / April 2021
    Published online by Cambridge University Press:
    14 May 2021, pp. b1-b2
    Print publication:
    April 2021
    • Article
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  • Volume function and Mahler measure of exact polynomials

  • Part of
  • Antonin Guilloux, Julien Marché
  • Journal:
    Compositio Mathematica / Volume 157 / Issue 4 / April 2021
    Published online by Cambridge University Press:
    14 April 2021, pp. 809-834
    Print publication:
    April 2021

  • We study a class of two-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.

  • COM volume 157 issue 4 Cover and Front matter

  • Journal:
    Compositio Mathematica / Volume 157 / Issue 4 / April 2021
    Published online by Cambridge University Press:
    14 May 2021, pp. f1-f4
    Print publication:
    April 2021
    • Article
      • You have access
    • PDF
    • Export citation

Page 95 of 818