Goncharov’s conjecture linking Sah algebra cohomology to algebraic K-theory

Establish that for all integers i and n, the Hopf algebra cohomology H^i of the direct sum of reduced spherical scissors congruence groups ⊕_{n≥0} \widetilde{\mathcal{P}(S^{n−1})} (in degree n) is isomorphic to the positive eigenspace of complex conjugation acting on K_{2n−i}(\mathbb{C})^{(n)}, the weight-n piece of the Adams decomposition of algebraic K-theory.

Background

The paper recalls Sah’s Hopf algebra built from reduced spherical scissors congruence groups across all dimensions, often called the Sah algebra. This algebra has coproduct given by Dehn invariants and appears naturally in scissors congruence theory and its connections to K-theory.

Goncharov formulated a conjecture identifying the cohomology of this Hopf algebra (graded by dimension) with the positive eigenspace for complex conjugation on the weight-n part of algebraic K-theory of the complex numbers. Proving this would bridge scissors congruence invariants and deep structures in algebraic K-theory.