The Strong Suslin Reciprocity Law

TL;DR

This paper proves Goncharov's conjecture on the strong Suslin reciprocity law, extending Milnor K-theory and polylogarithmic complexes, with applications to hyperbolic polytope scissors congruences.

Contribution

It establishes the strong Suslin reciprocity law and introduces a homotopy invariance theorem for polylogarithmic motivic complex cohomology, confirming Goncharov's conjecture.

Findings

Proof of the strong Suslin reciprocity law.

Construction of hyperbolic polytopes from rational functions.

Explicit computation of hyperbolic volume and Dehn invariant.

Abstract

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.