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The Chern class for $K_3$ and the cyclic quantum dilogarithm

Kevin Hutchinson

Abstract

In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in their recent paper in Ann. Sci. École Normale Sup. that $R_ζ=c_ζ^2$, where $R_ζ$ is their map on $K_3$ defined using the cyclic quantum dilogarithm and $c_ζ$ is the Chern class map on $K_3$.

The Chern class for $K_3$ and the cyclic quantum dilogarithm

Abstract

In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in their recent paper in Ann. Sci. École Normale Sup. that , where is their map on defined using the cyclic quantum dilogarithm and is the Chern class map on .

Paper Structure

This paper contains 7 sections, 17 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Review of some background
  3. The homology of a cyclic group
  4. $K$-theory, the Hurewicz homomorphism and the Bloch group.
  5. $K$-theory with finite coefficients and Chern classes
  6. The element $\eta_\zeta$
  7. The Bott element in $K$-theory

Key Result

Lemma 2.1

Let $C$ be cyclic of order $N$ with generator $t$.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 19 more