Hilbert's third problem and a conjecture of Goncharov

TL;DR

This paper links Hilbert's third problem and Goncharov's conjecture to algebraic K-theory and Chern--Simons invariants, providing new insights into scissors congruence classes and Dehn invariants.

Contribution

It reduces the generalized Hilbert's third problem to the injectivity of Chern--Simons invariants and establishes a version of Goncharov's conjecture relating scissors congruence groups to algebraic K-theory.

Findings

Reduction of Hilbert's third problem to Chern--Simons invariants injectivity

Establishment of a Goncharov conjecture variant connecting scissors groups and K-theory

New framework linking geometric scissors congruence to algebraic invariants

Abstract

In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Chern--Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic $K$-theory of $\mathbf{C}$.