Hilbert's third problem and a conjecture of Goncharov

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 $\mathbf{C}$.